There are some good medium calibers available in levers. The pistol calibers are mild mannered for the most part, and with the right loads will do for deer etc. The 357 in a lever makes a good combination small game/deer round. The 44 mag covers even more ground, particulalry if you reload.
My own choice is the 30-30 for the time I spend off the mountain (not in grizzly country). I shoot grouse, squirrels and bunnies with a .310" diameter round ball load with 3 grs Unique powder. The noise level is about like a standard (not high velocity) 22 LR, and it doesnt ruin much meat. I also shoot a 120 gr cast fp bullet with 6 1/2 grs Unique to duplicate the 32-20 round, it also makes a good small game load, but makes more noise than the round ball loads. The round ball loads are great snake loads also.
When using either of these loads, I leave the magazine of my Winchester 94 loaded with full power 170 gr stuff, and just open the action without raising the lifter and drop the light loads in the chamber, shoot (or not), and eject without raising the lifter, or just push the lifter back down to leave an empty chamber or load another light load.
If you arent set up to relaod, you can get a traditional Lee Loader for under $20. If you don't want to start reloading, you can get a chamber adapter to use 32 auto rounds in a 30-30. They work well for small game etc.
You can load a big gun (relatively speaking) down like a small gun, but a small gun will always be a small gun.