.22 LR? It's taken everything.
All around, right? Rabbit, squirrel, woodchuck, raccoon don't need a .30-06. There's only a few species that need 1000 foot pounds of power out to 400m. Deer, bear, hog, antelope, sheep, and elk are about it. That is actually a limited group of trophy animals, and they won't put much meat in the pot on the annual basis. Like as not, small game is the major source.
That means you want something a lot smaller for the majority of the actual game you will be shooting, with enough power to take down the rare and occasional larger animal you might stumble across. Since the .270 bullet can do that, put it in a case short enough to feed from a magazine. Self loading guns are far superior to getting shots off a small quarry moving through ground cover, and intermediate calibers don't punish the shooter, use up as much powder, and have quicker sight realignment after a shot.
The recommendation I'm making will get no appreciation in this thread, but here it is: best ALL AROUND for North America would be an AR15 in 6.8SPC.
Tried the .30-06 scoped bolt, tried the .30-30 lever (my Second choice,) moving to the future, not living in the past.
I have no problem with someone suggesting the 6.5 Grendel, but the OP said no 600m shots, right?