I've used the .308 in a Belgian Browning BLR for literally everything from field mice to Shirras moose. I used it in an M14 over the course and at a thousand in competition.
The .308 and the '06 are twins with bullet weights up to 150 gr., 165 and up, the '06 gets a velocity edge, but not necessarily the inherent accuracy that the .308 case gets in most rifles. I've had 'bum' .308's, and exceptional '06's.
Long range starts around 600 yards. It helps to shoot a lot, have a knowledgeable spotter, and be able to dope the conditions. What you use isn't near as important as knowing how to compensate for the wind, mirage, temp, humidity, and light, and do it while you're shooting. The only way to learn is practice, and GOOD practice, not just blowing rounds into the berm.
Right now, the hot setup seems to revolve around one of the short, fat cased, 6.5 bores. The .260 with load development should be a humdinger, and easier on the shoulder than a lot of other rounds, especially if you want to do it in a rifle as light as a Handi......
Good luck, and have fun!