The rule in alaska is "no rimfire cartridges, for large game." So *theoretically* (I'm sure kieth wouldn't advise this
) You can hunt grizzly and moose, with say, a .25 or .32 auto. In these cases, extreme judgement is advised.
That out of the way, let me tell you a .22-250. story. My dad had coyote hunted for quite a while with his .22-250, when they decided to go on an antelope hunt. He had recently bought his Winchester .270, but hadn't hunted with it, only shot it at the range. and sighted it for 100 yards. He spotted an antelope at about 400 yards, just standing there, but didn't know how his .270 would make the shot, but he did know his 2-2-2-5-Ohhh as he calls it. He benched with that, and dropped the antelope cold.
That was over 15 years ago, and now he says he could definatly do it with the 270, but he wasn't familiar with it, and didn't want to risk it.
I think the bottom line is, what kinda shots, what kinda country/cover/and what kind of a shot you are. A 22-250 bullet is small and fast, it can blow-up, or deflect on a piece of grass, or stick. If you get a jump shot and hit it in a leg, the bullet might blow up in the meat, and never reach vitals or bone, you'll be dealing with a wounded animal, that is probably coyote food. But for standing shots, heart/lung, that you feel you can make, it will work at almost any range (past 500 .22's start to loose the "blow-up" effect, and basically it's now a .22 rimfire.). If I could onle take one, I'd go with the .270 and load up/down. Good luck.