Just for giggles, I looked up factory (SAAMI approved) rounds, based off the .30-06 case (.270 Win bolt-face, which is the rifle in question) and came up with:
.25-06 Rem
6.5-06 A-Square (SAAMI approved 1997)
.270 Win
.280 Rem
.280 Ackley Improved (SAAMI approved 2008)
.338-06 A-Square (SAAMI approved 1998)
.35 Whelen
Since the goal was elk, all these would work, but since .270 was the baseline, then the .280 AI would be a great choice and a bump up in power. The .338-06 A-Square would be a sizeable jump in power. The .35 Whelen is a golden oldie that would certainly do the trick, too, for any North American game. Tons of wildcats out there, but being able to get factory ammo would be a plus (I'm assuming.)
Since I have a soft spot for the .270 Win, I'd suggest a rebarrel in that caliber again with a 24" barrel and a 1 in 8 twist so heavier for caliber bullets could be used.
I'm actually hoping you find that the current gun just needs a good, thorough cleaning to 'shoot' well again. That would be the cheapest solution, and the .270 Win still works fine for elk 'as is'.