243Win,
250 Savage (AKA 250 -3000)
257 Roberts (7X57 necked to 257 Main reason why no one has made a factory 308 case necked to .257)
25-06 (30-06 necket to 257)
25 WSSM - win super short douplicates 25-06.
257 Wby Mag (264 Win Mag Necked to 257 with Wby neck) Yes I know it is a 300 H&H cut back and necked but it is a standard action lenght case. New guns by Wby(Vanguard) or Howa are available for under $400. It is a Lazer but ammo and cases are expensive
2 friends in MT and one in ID swear by this as a do all from preditors to elk.
260 Rem (308 necket to 264/ 6.5mm)
6.5X55 Swede
264 Win mag. (338 Win mag necked to 6.5mm)
7mm-08 (either 308 necked down or 243 necked up to 7mm
284 Win
280 (AKA 7mm Express - 30-06 necked to 7mm/284)
7mm Rem Mag (recoil will be the same or worse than the 308 or 30-06 you mentioned)
All these rounds have great sucess on Deer and have options for lighter varmint/ Preditor projectiles. But I would just use the same rounds you use for deer.