I've seen something similar before, and sometimes I think about doing it when I have no assistant :p
Vance, maybe that could be adapted for your hat! All sorts of hat for that matter ;)
One problem though, bouncing light off an umbrella with a hotshoe flash is not gonna have too much reach. I tried bouncing off from a Metz 70MZ-5 and the end result wasn't that bright, albeit soft enough.
Hmm...nice hat..a new trend for keeping the sun off and taking picts. There used to be a kind of Frame affair that connected to your camera and had a big flat umberella on it. I was a department store photographer back in the mid 70s and used to walk around and photograph different appliances like toasters and stoves then build advertising. nice flat - neutral lighting. cumbersome to use, but nice to have. The "hat" seems to be much the same lighting.