There are a number of obvious causes for this.

1. Scarcity. The more an animal is hunted, the fewer there will be. By the late 1800's, larger game animals like deer and turkey had dropped to very small numbers in the east, while buffalo had all but disappeared in the west. Market hunters ( like the young Annie Oakley) were still reaping huge numbers of wild fowl for the fashion and restaurant trade, but those were also seriously declining. At the same time, wild lands for habitat were also shrinking, being logged or farmed. When Teddy Roosevelt managed to get market hunting banned, some of those populations came back- (like whitetail deer, now often a suburban pest). But in many places, game animals just weren't abundant.
2. Practicality. In wilder regions (like the Appalachians) that had enough habitat to support game populations, animals like possum, groundhog and squirrel would still be hunted. Uncle Dave Macon sang at least a couple songs about hunting groundhogs. But although squirrel stew would be considered a common cure for someone with a cold in the Smokey Mountains there's not much meat on a squirrel and it's a lot of work to skin a few of them for a dinner. For meat, the hillbillies would turn their hogs loose in the woods and let them root for acorns and (until the Blight) chestnuts. One large hog could give them around 200 pounds of meat. That's a lot of squirrels and groundhogs. And a hog could be smoked and salted. I don't think anyone has ever bothered, or tried, to salt a squirrel.
3. Production and transportation. Once refrigerated trains were available, it was possible to raise livestock on a large scale in places like Texas, slaughter and process them in large numbers in some other distant place like Chicago, and ship them to places yet more distant like New York. That kind of economy of scale tends to simplify, create big suppliers for big markets. It was easier to have large scale production of cattle and have a large market for them. It's comparatively harder to raise a lot of rabbits, more trouble to slaughter and process them, more trouble to find a place to sell them. A restaurant will have to work a little harder to find a supplier for rabbit for its menu, but it can simply order beef from a butcher.
4. Cultural bias in taste. When the German hunter Friedrich Gerstaeker was traveling in the Arkansas territory in the 1840's, he was early on invited to sit down with a homesteading couple and share their dinner. He sat and watched the wife prepare a possum. To him, it seemed as though she was cooking a very large rat, and he avoided eating it. Some cultures just don't have adventurous tastes. Plenty of people will tell you rattlesnake and alligator are pretty good eating. But you won't find them in too many grocery stores.

I can't really quantify the order, here. And there's one more that could also be important. The later 19th c. was a time when people were conscious of their social status, and this could have been another marker. A poor hillbilly was going to have to be scrounging in the woods for squirrels and possums, someone better off had chickens in the yard and hams in the larder.