Bardic Love,

I have a sunhat made with cotton twill. it's VERY gentle and soft, and the hat has a beautiful shape. The closest thing out there is this lady's work. She makes hats full time for a living.  http://www.gottahavahat.com/adult/index.html

Here's the discontinued one, just so you have an idea of what I mean. It's from Vermont Country Store and is item #42142. Here's some alternative views.
http://www.vermontcountrystore.com/shopping/product/alternateviews.jsp?iProductID=10649&picindex=2
This hat is not available anymore, but hats of its kind are everywhere. And these cotton hats are very hair friendly. :D

ETA Happy hat shopping!

Anais

Thank you Anais, these look really good. Maybe I'll be able to look into getting one or find something similar now that I have something to go off of. Hats like that would be perfect for keeping the sun off me while I'm at the beach.