I'd also echo going to a .243 for those longer shots on coyotes at 300 yards or more. An all-around varmint scope is a 4.5-14x Mueller. I just got one and feel it's perfect for all the varmint hunting I do. Much more power and it becomes a bench-only rifle.
Years ago, I used to love the 6mm Rem, but found the rifle too heavy to lug around the 'chuck fields. Now I have a .243 Win and it's fantastic. It is also very good on deer.
Group tightness doesn't mean as much for long range hunting as reduced wind deflection. Wind is more difficult to judge than distance/trajectory. A rifle that groups 1 MOA can kill a lot of varmints out to 400 yards, provided the bullet's wind deflection is relatively low. The 85-90 grain 6mm boattails are great in the wind. Better .243s can shoot 1/2" groups, but can hit long-range varmints way better than a .223.
Recoil with the .243 Win is mild, but barrels heat up faster than the .223 Rem. That's not a problem for most hunters.