If your just shooting coyote for fur, I would stay away from the .243 and go for a 17 hmr or .204 Ruger. the 243 leaves really big holes in them. Great deer Calibre.
I started off with a 30-06. I've reloaded my own loads which were fun but then realized.. its too much. It left fist size holes in my deer and obliterated coyotes.
I bought a .243. Love it! My fiance and I both shot a whitetail with it this year and doesn't leave fist-size holes in the deer. I shot a coyote with the .243 and totally blew a huge hole in it(fist size). Well.. does it kill coyotes.. oh yea. But it does not help if you're trying to save the fur for auction(when they inspect it, they decrease its value to almost nothing). So basically, If you want to make money and skin, flesh and sell the fur at auction go with a .204 ruger(rounds are 33.99 for 20, Hornady vmax 32-40gr. I'd suggest to get into reloading since it will be cheaper in the long run). If you just want to kill them any calibre will do. .243 rounds range from 26.00 a box. Can't go wrong. Hope my story and experience helps. BTW if your planning to skin and make money off the fur make sure you inform on the regulations and laws in your province. In saskatchewan you need your hunter safety(35.00 online with challenging the test for 20.00 with a qualified intructor, or 150.00 in class session. You need this in order to get your trappers education course.), trappers education course(30.00). It is against the law in saskatchewan to transport any fur bearing animals including coyotes so literally you shoot them and leave them. If you get caught transporting them here. It's huge fines with the possibility of losing your guns, truck and anything else that was used in the shooting of the coyote. So not worth it in my opinion. But to eaches own. Good luck!