Your current 270 is fine. Now I am a looney and have all 3 of the rounds the OP mentioned, along with a few more, like I said I am a rifle looney!
If I was looking to compliment a 270 and was considering the OP's choices of the 06 or the 300, I would go for the 300. The 300 can do everything the 06 can do and more and can easily be loaded down to 06 performance and recoil level, but the opposite is not true, the 06 cannot produce 300 ballistics.
The 06 is a GREAT round no doubt, but just is not that much of a step above the 270. The 300 can provide a worthwhile bump IMO.
