I wear my beloved Sunday Afternoons Adventure Hat for both sun and rain. It looks dorky--it makes this short, stout granny look like a mushroom--but it shades my whole face so I don't have to use sunscreen except around water, snow, and light colored ground at high altitude. This is a bonus for me because I'm allergic to most sunscreens.

The hood of my rain jacket is big enough to put over it so that if it's pouring, the hat doesn't leak through its ventilation system. You could achieve the same thing with something like a shower cap or a bowl cover (do they still make those?) that would cover the mesh around the headband.