The first question is how much weight are you willing to dedicate to your firearm. Decide on a number and be sure to include everything associated with the gun that you would not be carrying otherwise. This would include tools, cleaning gear and extra belts, slings and pouches.
The second question is will you be shooting big game or not. One does not shoot 100 lbs of meat just to extend your stay for a few days. That is two to three weeks worth of food for one. If you intend to extend your stay by weeks then bring a deer rifle and call it a hunting trip. Avoid any kind of compromise gun. Otherwise think about small game up to the size of porcupine or beaver.
I would consider fishing tackle and a light-weight small bore rifle. The .22 magnum rimfire is my personal favorite but plenty of folks use the .22 long rifle as a primary foraging gun.
If you are a realy good pistol shot then an accurate long barreled handgun will save a bunch of weight. You should be able to manage at minimum 3 inch groups at 20 yards before you depend on taking game with it. A .22M, .32 or .38 special would be my choices. The .22 lr in a handgun just doesn't have the oomph I want for larger small game.
Leo