You need something between .243 and .270. I like the 6.5 Creedmoor and .25-06 ideas already mentioned. Alot of other great choices in that range. .257 roberts, 6.5x55 swede, .260 remington, .257 Weatherby Mag to name a few. Cant go wrong with a .270 either but thats getting close to your .30-06 in alot of ways. 7mm-08, although a .284 diameter is getting closer to .30 would be a good gap filler for you since it is short action unlike your long action 30-06.
Blessed be the Lord my strength which teacheth my hands to war, and my fingers to fight -Psalm 144:1, KJV, a Psalm of David