Just how wide is the brim? I find that with 3" or wider, the drainage from the brim goes well enough back to drain onto my coat rather than inside it.
Tried a cowboy type hat when i was over n the states and a Australian type drovers type hat, both gave me a VERY sweaty head and both tended to drip in heavy rain.
I found when i tiled my head forward the rain clung to the underside of the brim and straight down my neck, brrrrrrrr.
As i say i'm not much of a hat fan anyways but i do prefer a decent waterproof jacket with integral hood rather than a hat when it's raining.