Obviously there is a lot more to this than expecting your handgun to make up for the inadequcies of your rifle.
Here in where I live, big game hunting involves more than waking up and walking or driving a short distance to my hunting spot. I hunt several hundred miles away, and this is out in the middle of nowhere. I am usually camping by myself. I have been out for 14 days at a time. After hunting I put the rifle in the truck since I sleep in a single man tent. When I am sitting around the campfire, walking around looking for firewood, taking a crap or whatever, it is nice to have a handgun on my belt.
When I am driving around back roads to and from the area I intend to hunt I also like to have a handgun on my belt and my rifle unloaded and cased.
If I am on an ATV, I have my rifle unloaded and cased and a loaded handgun on my hip.
If I need to finish off a game animal, I would much prefer shooting it at close range with a handgun rather than putting a .30-378 into it.
If a good close range opportunity presents itself, I will transition to my handgun just for the opportunity to take a head of big game with a handgun. This has never happened, but it might someday. Of course it has happened when I didn't have a handgun with me because the law didn't allow it.
I have used my handgun to bust a coyote or a jackrabbit. I could have used my rifle, but felt the handgun would be more fun.
I buy these handguns and like to carry them even if they had no real purpose.
I have also carried light loads such as .38 Special wadcutters loaded to target velocities for small game camp meat. Shooting a cottontail or a chucker with a centerfire rifle wouldn't be a real good move.
I have told this story several times, but I used my handgun two years ago when deer hunting. I shot a nice mule deer buck at about 50 yards with my .30-06 (150 grain bullet at 3000 fps). As soon as the trigger broke the deer vanished. It was standing with a group of four other deer. They were still standing there, but he was gone. Knowing I made a good shot I walked about halfway to the where I last saw the deer. I took off my pack, took off my coat, sat down my rifle, rolled up my sleeves, got out my knife and some paper towels.................... I walked up to the deer at which point it jumped up and began running after the other deer at top speed. I drew a Ruger P89 from a shoulder holster and shot the deer twice bringing it down once and for all. Now I know that the deer would have went down eventually anyway. In fact, in hindsight if the deer would have run about 200 yards I could have driven the truck right to it. But, I did use my handgun and it was pretty cool.
On the other hand I went down to Arizona, right on the Mexican Border southeast of Tuscon. It was HAM (handgun, archery, muzzleloader) season for javelina. I took my muzzleloader and my Ruger Super Blackhawk on my belt. Either gun was legal to take an animal. Right off the bat I missed a very close, easy, broadside shot with the muzzleloader
The javelina just stood there for perhaps five seconds. I thought to myself, there is no way it is going to stand there and wait for me to reload. Eventually it ran off. 10 minutes later I was kicking myself in the behind because I forgot I had the handgun and might have got off another shot if only I didn't have my head in the wrong place. As I continued to hunt I realized that the desert was FULL of illegals running the border. In a couple hours I probably saw a hundred of them. I was then VERY happy that I had a handgun on my hip instead of only a single shot muzzleloader.