I wear hats for most outdoor activities. It's just a habit I picked up in the military. I do have a couple of fedoras and a Panama, several baseball type caps, some wide-brim hats and long-bill caps with neck drapes, etc. I tend to choose the appropriate hat for whatever form/function I am engaged in.
As an example, I like the Kangol for it's packability, crushability, and quick access. As an example I recently rode from Houston to Saltillo (Mexico) on my motorcycle. The Kangol is perfect for this once the helmet comes off. It stores easily into limited packing, and covers my "helmet hair" when the helmet comes off. The Kangol is mesh, so very breathable as well.
All I need is a bota bag to complete the picture
Oh! Almost forgot. Tilley hats are great. I have a Tilley-like hat with a neck drape that I use for my Africa work.
Cheers,
M8