In keeping with the title of this thread, I shoot a Savage 12FV in 22-250. For coyotes, I think it can't be beat. However, I'd also love to have a .243 (or 6mm Rem) for a windy day coyote rifle.
When I got the 22-250, I was using Sierra 55gr BlitzKings, but it seemed that every other coyote was a splash-spin-run. The BlitzKings are just too fragile for coyotes out of the 22-250. They might be fine in a .223, but I think the 22-250 just pushes them too hard until you get out to the longer ranges.
So, since most of the coyotes around here tend to be on the mangy side and the hides are worthless, I started loading the Sierra 65gr SBT in the 22-250. I think that with the BC of that bullet and the velocity that the 22-250 can drive it at, it's biting at the heels of the .243, and so far, I've started to lose my lust for a .24 caliber. Every single hit with that bullet so far has been a bang-flop. No spinners, just DRT (dead right there). Should make a decent deer load too. So, if you've already got a 22-250, try the Sierra 65gr SBT's if you've got a 12 twist or faster. Any slower (14 twist) and I don't know if the bullet will stabilize. The box says 1 in 8 or 9 or something like that, but I know a 12 twist will stabilize it out to 350 yards (the furthest I can shoot here at home). I think with that bullet the gap between the 22-250 and the .243 narrows quite a bit.