Lightweight non-cotton baseball-cap style hat?

[ezNomad]

I like that Nike lightweight training cap too and have one I use regularly. The material is ultrathin, great for warm/wet weather. However, it's not a good insulator for really cold weather. For that I recommend a light beanie-style hat that can be pulled down to protect the ears from cold. I have a light soft fleece one bought for $5 that also can be stuffed with clothes to make a pillow at night.

[BigHodag]

Academy sports carries a line of lightweight, nylon fishing hats that may fit your need. Their cap features a fold down sun screen to protect one's neck and head. There's also a "boonie" style hat. Prices are about $12-14.

[swjohnsey]

I got a Levi baseball hat for Christmas, 100% polyester, very well made, very light.