I was raised in West Texas. They're just hats. Though, you might take a clue from WriteKnight and be descriptive of the hat. Some folks wear broad hats made by Stetson. Some wear tight, weathered and sweat stained straw hats they got at the dollar store. Some men just wear a ball cap if your writing about current ranch work, your ears get burned so I wouldn't recommend it. And then you have to consider the material. Felt has many grades that have widely different price ranges. You can get a felt hat from WalMart for $20, but the felt is cheap. You can get a beaver felt hat for more than $500, a lot more if you shop at the right place. Straw hats aren't all equal either. There's straw and palm and reed.
If I were writing a cowboy hat, I'd describe the hat and then just refer to as a possession of the wearer from then on, if I referred to it at all.