The .270 and the .30-06 are the same case, the difference in recoil will be because of bullet weight. Same bullet in both rounds will have the same recoil.
The same can be said for all the cartridges in the .308 family, .243, .260 Rem. 7mm-08, .308, .338 Fed - the recoil difference is a function of bullet weight.
If you have an 06 for the big stuff, what's the small stuff you want to step down to with less recoil? Deer? Varmints? In other words, aside from less recoil what do you want the other rifle to do? Shoot flat on coyotes and smaller? Be a closer range deer gun that is fast and easy to handle? If the latter you might think .30-30. If the former a .223 may do well. If both then a .243 can do it, but a .260 may do better.