The iconic Australian hat is called an akubra and is made of rabbit fur. (Sorry, rabbit lovers, but we have way too many of them here.)
Your super-waterproof version, a non-amos, may be made of oilskin, like the equally iconic Australian coat, the drizabone.
Actually, I think they are both trademarks and need capital letters, so I don't suppose they'll be turning up as common words anytime soon.