Seriously, your 223, 260 and 30-06 would work just fine. Save the 338 for those assault coyotes
If you hand load, there are plenty of choices for your 260 and 06. I think Remington is still making the accelerator rounds for your 30. I have tried them in my 308 and they group about as well as standard factory ammo, but I prefer to handload some 125-130 grain stuff for varmints and take advantage of the smaller groups my rifle is capable of. I take my Remington VS in 308 out with my favorite handloads, as well as my 10/22 for the close in stuff, loaded up with 40 grain solids, and it has worked very well. Also have a 270 win and with 100 grain handloads, it reaches out as far as I have any business shooting. If there's a lot to shoot at, rotate your rifles so you don't overheat them. I also have an M96 Swede which I have purposely left stock and open sighted and try my hand at the running stuff, which is coming along nicely.