There is no denying it; the wool felt fedora and akubra style hats are huge at the moment, whether it be winter or summer in your corner of the world.
Being wool/felt means that these hats can keep you positively cozy in the colder months, but keep you sunsmart in the warmer months. Basically, invest in a good one and you're pretty much set until the thing falls apart.
I myself just purchased (and very promptly received) my very own Lack of Color 'Montana Midnight Muse' hat, which I am completely obsessed with and will probably only take off to shower. Be warned though, this is a serious hat and you may have to tip your head sideways to fit through doorways.