Hat for hot sun…

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Outdoor Research Sun Runner Cap

Soft, breathable Supplex® nylon fabric is lightweight, quick-drying and offers a UPF 30+ rated protection from harmful UV rays
Removable cape offers full protection to your neck and face
Stiffened bill offers sun protection for your eyes and face
CoolMax® polyester inner headband wicks away moisture to keep sweat out of your eyes
Mesh side panels allow ventilation; mesh panels can be covered by the cap for added sun protection
Hook-and-loop? adjustable back band and adjustable cinch-cord provide a comfortable, secure fit
Sizing information for head circumference is as follows:

Small fits 21-5/8 in.
Medium fits 22-1/2 in.
Large fits 23-1/4 in.
X Large fits 24 in.
Imported.

Item 721671

I sweat even when I shouldn't. I recently purchased a Columbia Coolhead cap that impressed me on a 13 mile, 90 degree, 100% humidity day hike. The built in sweat band did a great job keeping the sweat off of my brow. It looks good too!

http://www.columbia.com/Men%27s-Coolhead%E2%84%A2-Ballcap-III/CM9484,default,pd.html

-Rudy

http://www.tinydeal.com/solar-powered-cooling-fan-hat-p-13645.html

Inexpensive too…

I got turned onto Tilley hats from a sailing friend of mine, and have never been happier.

http://www.tilley.com/canada_en/shop_men?cat=44&t_filter_category=206

The airflow hats are fantastic. Just MAKE SURE you get the hat a couple sizes larger than your measured head size (they have very specific instructions on this). It is designed to fit very loosely, but will not blow away due to the front/rear head straps. I am always surprised at how comfy I feel, even in extremely hot & humid weather. The loose fit makes a huge difference with this feel, in my opinion.

The Sunday Afternoon Sport/Water/Adventure hats are quite light. My sport hat is 2.2 oz. They aren't "miraculously cooler", but do have mesh panels that allow a little air flow. I think they meet your other criteria except the wicked look part.

On the Tilleys, taking out the flotation foam thingy inside the "hidden compartment" seems to help with the coolness factor as well.

X2 on the OR Sunrunner. I love mine. The added neck cape thing makes me look like a dork, but it works great. I also find the neck cape helps keep the wind off and helps keep mosquitos from biting me. Good side benefits. Works well in combination with my Patagonia Sun Hoody, too. Lifetime warranty, too! Get the white one for maximum sun protection.

I like my REI all synthetic hat with mesh side vents. Here in the desert southwest its a great shade hat. I've slipped a piece of that thin, white closed cell foam (used in shipping) in the crown for a bit more insulation on very hot days.

It's lighter than my cotton wide brimmed had and dries FAR faster.

Patagonia vented spoonbill

This is a great hat. The drape doesnt fall on your neck and irritate you, it stays kind of suspended in air. Its thick. The bill is really long and shades well. Not quite as dorky looking as the sunday afternoons sun hat, but similar protection.

Downside is the hat is kind of heavy.

The thing I like about the Sunrunner is the ability to remove the shroud and use it as a ballcap in camp and in town. I have a Walmart knockoff with a sewn in shroud, but I much prefer the Sunrunner.

another +2 on the Sunrunner. I usually wear it without the shroud when hiking, then attach it when I'm out in the sun for any length of time. That fact that it's totally removable, rather than tucking up inside and affecting the fit and coolness of the hat, is good.

I also like Tilley hats. Depending on where you hike, a cotton or hemp Tilley can be used as a swamp cooler – just soak it with water and the evaporation will cool your noggin. The provide a lot of shade, though not as much protection as the Sunrunner.

"I have a big, bald head and sweat like a pig."
Well, me too. I have the OR Sun Runner that people here recommend. I love that hat, but find it a bit hot in the humid summers here. I mostly wear it in spring, fall, winter (over a balaclava). I do really like the neck gaiter – almost serves like a hood in sun and wind protection.

As an alternative I've looked at runners caps. There are other running caps that have a similar design, but the coolest one I have tried up till now is the Salomon XA + CAP II. Also has a neck flap, but it's made from thinner material, with a bit more venting. This makes the Salomon cap much cooler than the OR Sun Runner (at the cost of some side protection).

ProTech. Very well ventilated.not super durable.

http://www.outdoorresearch.com/en/or-gear/hats/sombriolet-sun-hat.html

This hat is only slightly less dorky than a sunday afternoons job, but completely eliminates any need for sunscreen, and is plenty light. I have had the brim "flip up" on me in winds, but it never goes anywhere.

I've worn it in both hot, humid midwestern summers and out west in the desert and it does fine in both, definitely keeps me cooler than heats me up. With the vents on the crown, it's not completely waterproof but the wide brim keeps the unpleasant rain off your face, and the DWR finish keeps it dry enough.

My one complaint is that filling the extension collar on my pack cause the long rear brim to rub. Manageable for a day or two (until I eat the extra height down :) ), but I'd say it's not really compatible if you use a tall pack.

I LOVE this hat, but it's not the most aesthetically pleasing thing around. It has great coverage to keep the sun off. The part that goes around your head is essentially an adjustable sweat band, with mesh above connecting to the hat part. The awesome part of this is that you get air movement all around your head, keeping you cooler, while still getting 360 degree sun protection. All for 4.6 oz! They're about $20 on Amazon. Also, Joe at Zpacks has a slightly larger version that relies on a visor-style cap in addition, but is waterproof and takes down to be flat (and double as a sitting mat!).

Edit: Crap, I didn't see the part about fitting under a hood. No way with this thing-but a hood fits underneath it, no problem! And it has a "stampede string" to keep it on in high wind. I've worn it in sustained 40+ mph winds with no issue.

The Sunday Afternoons Adventure hat is good for sun protection and works well with a backpack with the "cape" style back panel that doesn't bump a tall pack. Looks dorky enough to be considered effective birth control. I guess you could wear it with a hood, but not my first choice there. SA makes some more traditional looking wide brim hats and some that are more like Tilleys at much lowe cast. None are hood friendly.

The OR Sunrunner is good. You can take your favorite ball cap and add a bandana by just tying two corners together with a bit of line and drop that over the top. Some simply put the bandana on their head and plunk the hat on top. You can soak the bandana for cooling too. For that matter, you could just wear a light colored bandana pirate style. Arrrrrr!

I like the Nike Dry Fit caps in white for a general purpose billed cap. They make some with a lot of mesh panels am that are better for moisture/cooling. Running stores will have a bunch of very light billed caps with lots of vents.

Tilleys are great and look good, but they are expensive and not hood friendly.

Bucket style hats are more compact and give good ear coverage, but not as much for face and neck. OR makes a few.

I have thick dark hair and I don't find any hat to be cooler when I'm hiking hard uphill, but a wide-brimmed hat protects my face/neck/ears. I think an umbrella is the coolest option.

If you're going to be serious about keeping the sun off your face (and you should be while hiking at altitude) then you have to resign yourself to not looking fashionable. Baseball style caps look great but just don't do the job.

Sunday Afternoon hats are great–wide brim that's like having a little umbrella on your head. Great coverage at all angles.

Sun Precautions makes a drape style hat out of Solumbra fabric (100% spf) that has a huge drape. It's white, which helps keep it cool. The thing is, you can adjust the drape to cover more or less of your face, right up to over your nose. This is great during mosquito season.

I have tried everything under the …ahem…sun. Ball caps, bandanas, military boonies although I have not tried a Tilley. I finally tried the Henschel Breezer and find it works for me. Keeps me cool in the summer heat. I spray it with some waterproofing stuff and it does pretty good in the rain even with all the holes in the mesh so I rarely use the hood on my rain jacket. Brim is stiff enough that it does not flap in the wind and works great with a bug net.

That being said, although the company claims it is crushable, it certainly does not come out of the pack looking spiffy and new…but I can live with it.

…

Granted, I don't have to worry about much blistering sun hiking in New England, but for hot weather, I find the booney style hats to be way too hot. For a hot sweaty hike, I go with one of the mesh runners ball caps. All of the running/hiking/biking apparel brands sell them. Lightweight mesh ball cap, usually with an absorbent inner band. They soak up sweat like crazy, keep it out of your eyes, and evaporate quickly. They can be easily rinsed in a stream. I have several and rotate them through the washer regularly — with pretty much every load in the summer. Perfect under a hoody. They usually have some reflective bits. I treat them with permethrin to keep the bugs away.

I've had versions from Brooks and from Marmot. I try to buy them in bright colors so I can wear them hiking during hunting season and for double duty when out for a walk/jog on roads in the neighborhood. They are ugly, but highly functional in hot, sweaty weather.

Marmot Tilden hat

Brooks Nightlife Running hat (with broad stripes of reflective fabric)

I'm looking for a hat too, after leaving my French Foreign Legion type hat on a rock last Fall in Little Lakes Valley. Why would you be wearing a hat with the hood up on a raincoat? Sometimes it is still raining when the sun is out though. Thank you all for the suggestions.
Duane

"Why would you be wearing a hat with the hood up on a raincoat?"
Because the brim/peak of the hood isn't firm enough so you need the brim/peak of the cap to make it firm enough.

I have used the OR Seattle Sombrero on a number of occasions (in both hemispheres) and have been very pleased with its sun resistance, yes the brim does relocate itself in strong winds. But the hat will keep the sun off whilst also keeping my head cool as well the pertex rain cover will help to keep your head dry. Overall I am very happy with this offering from OR

Disclaimer: I am not paid by OR, however, only ever use OR hats.

"Why would you be wearing a hat with the hood up on a raincoat?"

Many a lousy hood design has been "cured" with a billed cap. The technique helps to keep from seeing the inside your hood when looking to the side too. Eyeglass wearers like the idea as well. Lo' and behold, a cap can even add a bit of warmth under a hood.