Sadly, I can’t help you with the beanie solution, but can say that I stick with a fedora. A stiff, substantial brim (unless you can pull off the floppy look, which I cannot) is the key.
Granted, I live in Austin so it’s rarely below the 40s. But a wool fedora will look chic and shouldn’t give you crazy hat hair once you remove it.
Thank you for the amusing images, and more importantly, I hope you find a solution!