I began seriously hunting coyotes about 15 years ago. I drug out my ole reliable Remington 700 varmint special in .22-250 and started calling. The first thing I realized was that, although great for groundhogs, a heavy varmint barrel wasn't the proper tool for predators. I sold it, and bought a Winchester model 70 featherweight classic in .22-250. To this day I feel that this is the perfect setup for hunting coyotes with a bolt action rifle. I had the model 70 at the gunsmith's shop getting the trigger lightened up, so I borrowed my uncle's Remington 700 BDL in .243 Winchester for a while. After dumping three or four coyotes with it DRT, I decided I needed a .243 chambered rifle. I found a 700 BDL custom deluxe on sale and quickly stepped up on it. When the 700 CDL came out I bought a new take off stock and restocked my BDL's shiny stock with it. This rifle served as my favorite coyote rifle for about 5 years or so. It seemed that I couldn't miss with it, and it's still at the top of my list as a perfect predator rifle. As I hunted harder and harder, I concluded that I needed an all weather rifle, so as not to goof up that polished steel and wood, so I added a Remington 700 SPS stainless synthetic in .22-250. This rifle seems to do it all for me. It has a sporter weight barrel so it's not too light for open field calling where I may get a 400 yard shot. It's also light enough that it swings easily on close in moving coyotes called from thickets. I've also added a 700 Mountain Guide stainless in .243 Winchester. It is a featherweight barreled rifle and carries and handles just like my Model 70, but is all weather. Over the years I've also used AR 15 platform rifles for coyotes. Liking a light quick handling rifle for coyote hunting, I preferred a 16" carbine AR. I built my first in .223 and had good luck with it. As I used it more and more I began seeing an issue with it. It seemed to me that the .223 from that 16" barrel was losing a lot of steam. I was having more spinners that needed a second shot than I ever did with my .22-250's and .243's. I shot a called coyote at about 75 yards square in the shoulder one morning and saw the 52 gr HP bullet impact. That coyote went down then came right back up and ran into the brush. I followed a light blood trail for about 100 yards then lost it. When the same thing happened again, I went back to my bolt action rifles in .22-250 and .243 Winchester. Because I really like the AR platform, I built an AR in 6.8 SPC. Shooting a 90 gr HP at 2850 FPS, it is an excellent rifle for called coyotes. Because of the poor BC of the short fat bullets though it doesn't shoot nearly as flat as the .243, but its really great out to about 200 yards. With the current availability of 6X6.8 AR barrels, I may make one of those my next build.
All in all, after hunting coyotes over the past 15 years, I've concluded a few things. First is that coyotes don't hold still for a shot like a prairie dog so your shot placement can be a little iffy. To CYA you need at least a good 55 grain hunting bullet at over 3200 FPS to reliably penetrate through the shoulders and into the vitals of large Midwestern coyotes. This is where a .22-250 rises above a .223. Secondly, a sporter weight rifle works best for both in close called coyotes and also for calling around large open fields where you may get shots at 300 yards or more. I, at least, shoot my sporter weight rifles noticeably better than I do my featherweight rifles at long range. An AR-15 platform rifle is a great choice so long as you can keep the velocity of your .223 over 3000 FPS or if you can use a heavier cartridge such as a 6.8 SPC or 6.5 Grendel, or better yet, one of those necked down to 6mm so it will shoot flatter. I've pretty much settled on .22-250 and .243 for bolt action rifles. What the .243 does with heavier bullets such as 85 gr Sierras, the .22-250 does with higher velocity with 55 gr sierras for me.