A bolt gun in .223 would serve well in the role you describe. My favorite coyote rifle is topped exactly as another poster already recommended: Burris FullField II in 4.5-14x with the Ballistic Plex reticle. Just about perfect, IMO.
As already stated, the .22-250 doesn't gain much over the .223 until you go beyond 300 yards, and even in the wide open sage country here, 300 yard shots and beyond are the exception rather than the rule.
If you're calling, you'll often have more than one dog come in at the same time, and sometimes from different directions altogether. I'm finding that it isn't altogether uncommon that I shoot at one dog, which causes another to bolt from where I didn't even see it. A repeater is the ticket for quick follow-up shots, though with some practice and the right rifle (preferably something in the Ruger No. 1 or High-Wall variety, not the break-action H&R or Rossi), one can reload and reengage pretty quickly.