My preferred method is to have someone else carry it for me. You asked!
Cant say for hunting mulies but I like my rifle at the ready when in the backcountry here in AK. Stuffed in a pack sleeve or somehow on the back of my pack doesn't not afford quick access. Its more comfortable so to say but I like my rifle slung for Bullwinkle and bou hunting. On the steeps in sheep and goat country, stuffed somewhere like a scabbard for easy carry.
The reason light is right for me or lightest is bestest. Don't think there is a perfect method, my recommendation, test them all and see what you prefer. Or course getting others opinions can help to start sorting it all out.
Edit, for moose and caribou I like a freighter pack with a frame so I can hook my rifle on the frame. For the steeps, internal frame pack with rifle stuffed in/on the pack body.
good luck