Your .243 is about perfect for what you describe. It is the classic varmint/deer crossover rifle.

I'd buy a "deer/light elk" rifle and forget about using anything else but your .243 on the coyotes and bobcats. A .30-'06 is one of the best ever, but don't leave out the .308 Win, .270 Win, .280 Rem, 7mm-08 Rem, .260 Rem, .25-'06, .257 Roberts, 6.5x55 Swede, 7x57 and .338 Federal, as well.

You really don't need a 7mm Rem Mag to hunt deer under 250 yards, but if you want one, go for it!