Before going to 243 for a dedicated coyote rifle, you might consider 6MM Remington. You get more case capacity, which translates into more velocity, and you will still be in a short action with the same bolt face. The longer necks make reloading and resizing a breeze. Talk to your gunsmith about twist rate to match the bullets you want to shoot. Brass is plentiful too. It will use most all of the same powders as 243.
After many years and several calibers, my hunting partner and I have both settled on custom made Remington 700s in 6mm Remington for dedicated coyote rifles. I am currently shooting 70 grain Bergers and he is shooting 75 grain V-Max. If you are going to go under 70 grains, you might as well build a 22-250 or 22-250AI. Going heavier, there is not much sense going over 75-80 grains for coyotes. Also keep in mind the 70-80 grain bullets have a higher BC and buck the wind a little better.
One last thought. If you go with 243 AI or 22-250 AI, give a lot of thought to your barrel twist rate. I have owned both calibers in fast twist barrels and was disappointed in barrel life and finicky loads. I also found some 22-250AI jacketed bullets blew up on me before reaching the target. There is lots of good info on this site about both AI calibers. It will pay to do your research. I am currently waiting on my gunsmith for a 22-250 AI 12 twist build.