For target shooting, I switch equally between the index and thumb button. The thumb button forces me to maintain strict tension and to always pull through smoothly so I don't slap that trigga' on the index release.
For hunting, I always go back to the index. It's always on my wrist whether I'm crawling on the ground or running around. Seems like more and more hunters are using handheld releases for hunting, however.
Depending on the release, you *may* need to adjust your draw length so just be mindful of that. Another thing I've noticed is that I tend to anchor consistently with the handheld release because I can always get the webbing between my 1st and 2nd knuckles right on my chin every time.