I've used my .243 when I first started out Coyote Hunting,it can put some serious damage on a Coyote and be pretty loud....if I'm Deer Hunting with the .243 a Coyote won't get a free pass or if I'm packing the 7mm-08 they are in deep trouble too but a Person needs to really designate a lighter Caliber Rifle for the Coyotes and Predators plus You might call in a Fox or Bobcat and a larger Caliber Rifle is going to do some serious hide/pelt damage!
As several mentioned the .223 is a great affordable round with many choices in lighter weight bullets that will be much easier on the Hides/Furs and will knock down Coyotes and be easier on the thinner skinned Predators too.I'm shooting Hornady 55 grain V-max bullets in my Savage Axis XP Camo .223 and could have went with even lighter bullets down to 40 grains which will push that bullet even faster!With the Synthetic Stock the Axis is a great all-weather Rifle that's light weight a ready to carry!
I would check into the newer Savage Axis II with a Scope Package that have the Accu-trigger,they are Factory bore sighted and shoot great with less time and Ammo to sight them in plus they are very affordable and accurate Rifles.One day You will want to save the Hides or like I do give them to a Local Trapper if the fur is in good condition or if You shoot a nice a Fox or Bobcat You might want to have Mounted or the hide Tanned to hang on the wall for display!
Once You get into Predator Hunting Your Goals may change and a *New Different Caliber is fun and enjoyable to try out.I really like My other Rifles I can choose from beside the .223....I have a CZ model 527 "Varmint" in a .204 and a Browning A-bolt Micro Hunter in a .22 Hornet for lighter thin skinned Game like Foxes,Bobcats and the occasional Groundhog!