My friend Keri broke her thumb at Alta a few years back. It broke in a few places and tore ligaments. She finished out the day like a trooper unable to hold her pole. The next day she went to the Doctor and had a cast put on. first thing she did after getting home with her cast was to see if it fit in her glove. It did not. So she cut the cast off and went to the store and bought a brace that would fit in her glove. Keri = HARDCORE.
If you're straightlining a chute or something sketchy and let go of a pole - hiking back up sucks. Put the strap between your palm and grip, not over the back of your hand - it gives you better control of the pole when using it and allows your hand to come off the top of the grip if it needs to.