Since you have a .308 you won't be swapping to 6.5G. Your choices are 6mm Creedmoor/6.5 Creedmoor or .260 if you are looking for relatively common with some factory ammo available. 6mm Creedmoor is almost imperceptibly better at close ranges and slightly worse at long ranges, for target shiiting but has very little factory ammo, has much less stopping and penetrating power, and a 30% reduction in barrel life. 6.5 Creedmoor is easier to seat the bullet to correct length than .260 when using the best combination of accuracy, BC, price and terminal ballistics, the 6.5 140 ELD-M. Hornady also has a factory 140 ELD-M loading that is a sub half MOA capable load.
6.5 Creedmoor has the flight ballistics of 300 Win Mag, but it a short action non magum load that has 2/3 the recoil of .308 you have now.