I’ve pondered and practiced this a lot, most often within the GCNP during Spring and Summer hikes back up from the Colorado River.
A basic $1 cotton bandana is the baseline in my mind. Cheap, multiple-purpose, can be sourced at any Walmart. I much prefer to cool myself by applying water externally than to sweat out that amount of water (and electrolytes) – it is less draining and, at the end of the day, I’m less sticky with dried sweat. Along the BA or NK, there are numerous spots to dip such a cloth into a stream and doing so reduces the amount of water I have to treat and drink. I drape the wet bandana over the back of my neck and THIS IS THE IMPORTANT TIP: flip it over every minute. The side against your skin gets warm, the exposed side gets cool, so keep flipping it over.
I’ve also used a strip of micro-fiber towels (I long ago ordered a big MF towel and have, for 15 years, been trimming off custom sizes for different purposes). The MF cloth dribbles less water down my chest which is neater, but sometimes, I want more of me wetted out. At least the MF can be squeezed nearly completely dry and put away when the sun goes down while one should stop wetting the cotton bandana somewhat in advance of cooler temps. The MF doesn’t seem to work quite as well for cooling my neck as the folded cotton bandana.
I’ve also used a men’s button-down collared cotton dress shirt (I set aside the lighter colored ones when retired from daily office wear). Water from the wetted bandana transfers to the cotton shirt and more of my upper body is getting that evaporative cooling. While wet cotton is lower SPF than dry cotton, it’s still far better than bare skin and the collar and long sleeves help minimize UV exposure and I haven’t had a problem with sun exposure through such wetted cotton clothes even through they’re pretty thin fabric. But, even more than the bandana, you need to stop wetting it out before the desert temps drop in the evening or you’ll have a damp shirt going into the evening. When I time it correctly, it gives me great cooling during the heat of the day and also functions a bit as a warmth layer in the evening / night.
I remove my hat for 10 seconds every few minutes to let the sweat evaporate off the top of my head and let the sweat band cool down.
As Jon mentions, dunking one’s head in a stream helps and it’s also nice for washing the salt off your skin and face.
I find it’s 10-15F cooler under a sunbrella and it is far superior UV protection than relying on sunscreens. I wear a LS fishing shirt with epaulettes that button and let me secure the sunbrella into one of the large cargo chest pockets and through an epaulette leaving it hands-free.