I've seen a roly poly pouch used as a water holder by fitting a twig to the bottom drain grommet. (
seanmullhall/silver fox on youtube)
You could probably do the same with a boonie or other style hat if the grommets are substantial enough. For example, I have a filson shelter cloth packer that this would probably work with. I hadn't really thought about it 'til I saw this thread topic.
I really like the sturdiness of the tin cloth. I have used it to gather tinder and fan a fire (too much a couple of times). I also frequently use it to shade things or keep things dry, either off the ground or from getting wet from rain. It is hot in the summer and doesn't breathe all that well. On the flip side, it may be the perfect hat for that 25 - 50 degree, cold & wet weather that we get in the spring, fall, & winter here in the midwest. It looks a little more woodsy than the folks around here are used to, so I'm trying to get used to the looks I get.
I have a lightweight boonie hat that I wore in the field for several years. Not waterproof at all, nor warm, but it kept the sun off great and you could use it for evaporative cooling or to put some ice on your head to cool down. The good & bad thing about it is that it doesn't really hold its shape, so you can stuff it well, but the brim will flop down in your face in the wind or water.
I wear a ball cap quite a bit as well, but I am generally concerned with getting skin cancer on the ears & head (since a number of outdoor guys i've known have had issues with this), so when I'm out for a long time, I try and wear a hat with more coverage.