You want a boonie type hat. The problem with a hat over a cap though is the brim length is usually too short. I always like at least 3 and a 1/4 to 3 and a 1/2 inches brim length.
If you want nylon and light, the Outdoor Research Sombrero will work perfectly.
A campfire will rape both of those.
There's also a Lawrence of Arabia baseball type cap by OR as well. Called the sun runner cap.
Akubra makes a bunch of longer brim felt hats.they will be much warmer and you will have to steam form the crown and brim, but they sell a huge Sombrero type hat.
Here's a vid:
https://youtu.be/vxDuDN7QakEI have tried a bunch of boonie hats in general, and most suck. The brim is too short like a usgi boonie or if longer it weighs too much and flops down at your face.
I did get one cloth hemp boonie to work although I can't remember the brand. Tilley th5 was too short of brim. Tilley th4 was too floppy and would form a droop at my eyes. The patagucci was pretty good, their hemp boonoe but the brim was just a bit too short for me. Usgi boonie and tilley packer were too short on brim length. I will dig around today and find the brand on mine. Fire season starts tomorrow anyways so need to grab it for station.
If you're a poor fag too, never underestimate a straw hat. You can't form it but many wagie marts carry one for nothing and they last long enough. You can find better designed ones on Amazon.
Just make sure your hat has a band so you don't lose it. You can easily add one in a straw hat by wrapping one around the front of the crown and poking the lace through the center on each side then tying it around your neck.
I have a 24 inch circumference head btw.