For many years the .30/06 was the Wimbledon Cup cartridge, until the advent of the .300 magnums. US military snipers used the cartridge with great success right up until the late 1960s in various versions of the 1903 Springfield, M-1D Garands, and M-70 target rifles, not to mention the M-17 sniper rifles issued with the earliest telescopic sights during the First World War. The most celebrated of these snipers was Gunnery Sargent Carlos Hathcock USMC, credited with 92 kills, made mostly with his M-70 .30/06 during the Vietnam War. Hathcock's battlefield zero was for 700 yards! That should cover the long range attributes of the .30/06.
When I decided I wanted to try long range shooting, it was with a .30/06 M-17 Enfield and a 3-9X40 hunting scope. I could keep 5 rounds on a 12" target at 850 yards on demand on a calm day with good light and no mirage. I was able to get hits on targets (18"X30") out to 1500 yards using sighting targets set on eskers high above my impact target. Yes I did lots of missing too, but I've often wondered what that rifle could have done with a modern long range scope.
There's a common belief that a .308 will be more accurate than a .30/06, but I believe that has more to do with the individual rifle than it does with the cartridge, and I doubt that any real difference in accuracy could be utilized in practical terms. There is also a belief that the .308 is a ballistic twin of the .30/06, but this does not apply to heavy for caliber bullets which when loaded in the .30/06 exceed the .308 ballistics by a significant margin, particularly if we ignore the 48000 psi pressure limit and load it to .308 pressures. My current target rifle is a .308, but from time to time I wish I had held out for a long action and a .30/06.