If you're hunting coyotes and mule deer, you're in the wonderful West!
And if thats ALL you would shoot with your rifle, mulies and coyotes and perhaps rock chucks and rodents, I'm surprised no one has praised and suggested what a fine choice a dual purpose rifle might look like in .243, 6mm Remington, .250 Savage and .257 Roberts.
However if elk might ever be in the future, thats a game changer! And the .270win would be an excellent choice in that case.
Don't overlook the .24 and .25 calibers for your original purpose, unless you need something bigger, in which case the .270, .308, and .30/06 are indeed responsible and good choices to kill elk with.
As for recoil, have you considered making sure to go to a synthetic stock? Besides the recoil management and absorbtion/flexing offered by a non-wood stock, synthetic stocks are basically weather-proof and "stable" in that they don't warp, and are often lighter weight too, when carrying looooong distabces!
Also I killed a couple cow elk with a .338WinMag in a M1500 Howa on a syntheitic stock and it did NOT feel any worse than an 06 or .270! Certainly felt a lot less than a walnut stocked M700 in .270.