I have a huge stack of hats for sun protection in just about every size and shape.
My hat of choice is a cheap wide brimmed straw hat. They are light, readily available in assorted styles and cheap.
The big downside is packability, you can’t do anything with it when its not on your head, plus when bushwacking it bumps on everything, but….
If I’m hiking in places where there’s no place to hide from the sun, they outperform everything else easily, so I put up with not being able to stuff it in my pack.
Use a head band in combination with a straw hat for sweat, and soak it with water for extra cooling.
I was in the Utah desert a couple of weeks ago hiking when it was 107F, the first day I used the OR Sun Runner hat, the second day I used the straw hat.
It was hotter the second day, but I was more comfortable wearing the straw hat.
I don’t typically have sweat running down my face or into my eyes, so my use of a wet headband or buff is just for cooling purposes.
At the trail head you get odd looks for wearing a straw hat, 5 miles out they’d buy it from you for big $$$$.
If sun protection was only needed for brief sections above the treeline, or something similar, I’d opt for something more like the OR hat.
I don’t have the Sunday Afternoon hat, but I do have several similar fabric wide brimmed hats. I find that once they’ve been crammed into a pack a few times they start to lose their shape, so I treat them carefully just like the straw hat.
Since the fabric and straw hats are treated in about the same way, I go with straw because it’s more comfortable and cheaper.
I find straw hats from $6-$20 just about anywhere.