I own 3 223s, 3 22-250s, 2 243s, 2 6mm Rem, and 3 204Rs. All have been used for shooting coyotes at one time or another. If I'm reaching into the safe to take a rifle out for the purpose of hunting coyotes 90% of the time it will be a 204 while half of that time being my Sako Varmint Laminate. The Browning gets chosen if I think conditions will be a little rougher and the Cooper comes out if I'm not walking too far or sitting in a blind over bait. I have shot and processed a lot of coyotes over the years and have found the 204 works the best for me when trying to save hides. I prefer to shoot coyotes at around 100yds if I can stop them there but I have also made quite a few shots in the 500yd range.
My second choice now would be a 223 over the 22-250 if saving hides was an option. Otherwise I would step up to something in the 6mm class over the 250. A 55gr NBT going 4000+ FPS out of my 6mm is pretty nasty on hides under 250yds. I still like my 22-250s but find they are coming out of the safe fewer times now that I have other options available.