I've used a few different handgun scopes for deer hunting, including a Tasco 1-4X, a T/C 2.5X, a Burris 1-4, a Burris 2-6, and a Bushnell 2-6. Other than for sighting in, the higher mag is not very useful. If it were me, after using all these other scopes, I would go for a 2X Leupold and not look back. The eye relief and small FOV of the higher mag optics on a handgun make them less than ideal for whitetail.
I guess the answer also depends on what type of cover you expect to hunt. If up close and personal, a good 3MOA dot scope would also be a good choice.
Handgun hunting is fun. I've filled a number of doe permits and taken a nice 6 pointer with my .44 mags, in ranges of 20-60 yds. Of all the handgun optics I've owned the Burris 2-6 is the nicest, and it RARELY leaves 2X...The Burris 1-4 suffered from too small a field of view, as did the T/C scope. The Tasco wasn't bad, but wasn't good in low light conditions.
Good luck with the healing AND hunting!