I've always been a strong proponent of carrying something in the woods.
I think the best all-rounder in the woods except in grizzly country is a .357 wheelgun. Since I do have largeish hogs and black bears in my area, I'd load it heavy -- 158gr or 180gr softpoints. If I'm carrying shoulder rig, I'll take a 6", if a belt rig, a 4".
If you absolutely want an autoloader, then pick one in 10mm or .45 Super. Bonus is, both are excellent manstoppers and can double as your EDC outside of the woods. Not to mention if you have one rigged for .45 Super, you can just swap out a lighter recoil spring and remove the buffer and shoot .45 ACP, as the cartridges are exactly the same size.
That being said, back when I had more than a couple of guns, my favorite woods gun was an S&W 629 4". 240-gr softpoints handled just about anything 2- or 4-legged that needed killing, and a few spare shotshells worked wonders on snakes or small game. The short barrel allowed for pretty good patterns and packability.
As for leverguns in .357 and .44 magnum: on deer sized game, the .357 guns are good to 100 yards, stretching to 125 if you know how it shoots well. The .44 is good to 150 yards, maybe 175 if you really know the gun. Otherwise, on that class animal, there's no real difference, both drop them quite well. With very large game, like moose, brown bear, etc, the .44 will have an advantage, although at that point, I'd rather have a .45-70.