Crossotter,
When my son got old enough to deer hunt with me, I let him carry the rifle and I carried a 44 magnum handgun, just for convenience. With a lot of practice over several years, I got proficient enough with the handgun that, when my son was old enough to hunt by himself, I continued to only carry the 44 magnum handgun. The gun was usually a 29/629 or a Ruger Redhawk, mostly loaded with a 250 gr. cast bullet of Keith design.
My years of handgun hunting lasted for about 20 years, and while successful sometimes and sometimes not, I found the 44 to be a very capable deer cartridge when used properly. I never shot a deer over about 60 yds. away and picked my shots carefully. I handgun hunted exclusively until the manufacturers started putting those fuzzy front sights on their firearms. Good luck.