I have to say that everyone is throwing out "ideal" cartridges for specific game but not mentioning the circumstances. This is like asking what lure I'm throwing without asking what the season, water temp, clarity, and conditions were. Are you saying you are picking a cartridge to hunt this animal in a game preserve in Tennessee which is completely forest and a 50 yard shot for a Dall sheep? Because you won't be using a .300 Win Mag. If you are hunting a Dall in their home, however, you will be using a .300 Win Mag. Are you baiting the coyotes? Because you can use a 5.56 or a 12 ga for that matter. Are you hunting them in open fields near my house after the beans have been harvested? You'll want a .22-250.
I'm just saying that the circumstances dictate your choice as much or more than the game. I don't want to know what lure you're using until I know the conditions your fishing in. Same with hunting.