Now that I look at it again I think I know what it is.
It is a common felt cowboy hat that has sagged and drooped with age, water exposure and wear. I'll bet you that if you took it to a shop that re-boxes cowboy hats they could return it back into an aproximation of its original turned up rim and western stylish self.
Personally I think it looks better how it is now. Sort of an sod-buster/mountain man look.
Assuming I'm right, I kinda hope I'm not because I only get to be right so many times a week, it should be easy to get one that looks like that.
1) Buy a high quality cowboy hat in an appropriate light shade. Buy quality because the cheap ones will fall apart instead of aging gracefully.
2) Wear and abuse it for fifty years taking care not to maintain it or its shape. When it gets so hats so nasty that it drops buzzards at 50' jump into a lake with it and wear it dry.
That would be the 'authentic' way to get one.
You might be able to get one faster by either buying the cowboy hat and selectively abusing it with water, steam and stomping it.
You might do even better to call a maker of high quality cowboy hats and have them send you one out that has just been felted to the general shape of a hat. But before it gets molded into the classic cowboy hat form. I once saw pictures of the hats before they are stretched, formed and finished and they kind of look like what Mr Kleb is wearing.
Remembering this is why it dawned on me that it is not a hat that was originally designed that way but rather a standard cowboy hat that has been regressed by wear and tear to an aproximation of its form before it passed the final stages of manufacturing a cowboy hat.