Not sure you're going to make it much lighter no matter what you do (other than changing the barrel). Long action required for 30.06 with a 22" barrel, rings, bases, and a scope usually equals about 9 pounds min. Remington does make a mountain rifle that starts at 6 5/8's pounds (no sights, 1/2 pound lighter than a BDL) in 30.06 with a 22" barrel and "slimmed down" stock. read that as very narrow like someone took a whitling knife to it. Add a scope, good base and rings and your up to almost 8 pounds. You could go Titanium, that'll save you another pound (Remington weighs in at 5 1/2 pounds naked). Savage has a 30.06 that comes in at 6 1/2 pounds with a synthetic stock and 7 pounds with a hardwood stock. Soooo, I guess you could save a 1/2 pound by switching to a hollow plastic stock. I'd take the stock off and weigh it, then compare it to the weight of a replacement, I don't think you're going to find a lot of difference. If the pound difference is going to be significant, I'd look for it in the other stuff I'm carrying, not the rifle. Boots come to mind, along with that really nice backpack my wife bought me for Christmas that weighs over 4 pounds EMPTY. It spends a lot of time in the truck.
Bill