Sounds like you would like to keep to the rifle's original history.
A new barrel should do that.
I'm not sure if a .30-06 will shoot as accurately as a .308 because the historical .30-06 bullet choices aren't as broad and, historically, most are probably hunting bullets or military issue, at least in my experience. Most of the traditional bullet choices probably will be heavier also.
But from a historical perspective, they will be true to the .30-06 heritage.
If you just want to use any 30 cal. bullets, there is no reason why the panoply of new, high BC bullets created for the .308s and newer variations wouldn't be worth a try. You may find that the powder choices might tend to slightly slower powders.
My .308s prefer H4895, IMR3064, Varget, N140, N540, and Reloader 15.
My 30-06s prefer IMR4350, H4350 for heavier bullets and some of the lighter bullets performed well with Varget and R-15.
You may have to experiment a bit to find out what bullet weights and seating depths the replacement barrel prefers, but it should be a fun process.
Good luck with the change and good shooting.