I got lots of suggestions. I have tried the stretching idea. I have had luck taking the band off hats made of straw or grass-like materials, wetting and stretching them. I have never been able to stretch a fabric hat enough to make it fit.
I plan to make a rain hat from the pattern one of these days. Synthia suggested cute oilcloth. That might be good, but perhaps a little too cute for my tastes. I am thinking more along the lines of GORE-TEX® . This is Oregon. We don't do cheerful rain gear. We blend into the landscape and are one with the rain. (ohmmmm) My friend Sheila directed me to Sunday Afternoons, a company in Southern Oregon. These look like very nifty hats and they do have an extra large size—which is not a guarantee that it will actually be large enough, but seems worth checking into.
For my hat I made my own pattern. It is a simple design, but required some math and figuring and trying. I constructed a paper hat and adjusted the sizing, then took it apart to use for the pattern. I am very happy to share my pattern with my big-headed sisters, if you'd like to make your own. I will need to draft it in Illustrator and make into a pdf, which I can email, so give me a week or two. I'm going to be gone for several days before I can even get to it.