If you can only have ONE gun that needs to be capable of pulling everything between coyotes up to caribou, then the .308win is a great choice.
Personally, I prefer to have multiple guns so each can be "perfectly" suited for its assignment, i.e. a designated coyote rifle, a designated bear rifle, a designated deer rifle, etc etc, but then again, I like variety.
Given your situation, I'd rather have a .223rem for coyotes, then a .300WSM for Caribou. Minimize the cost and pelt damage for smaller game, then bump up the power factor for the heavier game.
But then again, beware the man who only has one gun...