I've killed coyotes with my Savage Axis 22-250 and never lost one, but they tend to run off and die. When I use my Savage Predator 243, I never have a runner on coyotes. So, the 250 will get the job done, but the 243 does it with authority.

Plus, the 243 has more range
with the authority. If you're in hill country and don't have a concern about background, I'd go with the 243.
I reload for both.

22-250 uses a 55 gr. Hornady V-Max loaded at 3850 fps.
243 uses a 75 gr. Hornday V-Max loaded at 3400 fps.