I mostly use white terry cloth cleaning towels, the kind you can buy in packs of 20 or more. The terry cloth is nice and absorbent, they are light weight and the right size. Not too big and not too small. They are inexpensive, so if you lose one it's not a big deal (aside from not wanting to litter on the course). You have a bunch, so if you are playing every day, can just hang them up after the round and then throw them all in with the regular laundry.
Also, I've found that in the summer a sweat wicking beanie or "do rag" is far superior to any hat at mitigating head sweat. I'll bring a second one for the turn on the really hot and muggy days.