Here's a bit of history: the original match/sniper load for the .30-06 featured a 172- or 173-grain boattail. It was most famously loaded by Lake City Arsenal, and this load was used for most of the famous military sniper shooting with the '06, including Carlos Hathcock's legendary 800-yard shot of the Viet Cong general.
The modern equivalent of this bullet (even better, actually) is the Sierra 175-grain Matchking. The Sierra bullet is used (surprise, surprise) on U.S. long-range sniper ammunition in the .308. It has already been used successfully at ranges over 1,000 yards in Iraq.
Therefore, your first and obvious choice for precision shooting in the .30-06 would be the Sierra 175. A ton of other bullets will work, also, but the Sierra would be my first choice. The Hornady 178-grain AMAX would also be way up there, as would the Lapua 185-grain Scenar.
Actually, the Sierra, Nosler, Lapua or Hornady 155-grain match bullets would also be great. We Palma shooters use those bullets out of .308 match rifles all the way out to 1,000 yards. The advantage of these bullets is that they have less recoil than the heavier bullets, and you should have no problem zipping them out at 3,000 feet-per-second out of a decent .30-06. However, they do drift a bit more in the wind than the heavier bullets, and they don't need the fast 1-10" twist of the '06 to stabilize (though they work fine with 1-10" twist barrels).
Some people have mentioned the 168-grain Sierra or Hornady match bullet. This is the all-time classic bullet, designed for 300-metre shooting in the .308. Most experts agree that its range is limited to about 600 metres, though. As long as you stay within that range, it works fine, but its performance gets erratic by about 800. This is the reason why Sierra developed the 175. The shapes are different -- the 175 is considerably more aerodynamic.
The advantage of using the 168-grain bullet would be if you had a hunting load for the same rifle with a bullet like, say, the 168-grain Barnes Triple Shock or 165-grain Hornady Interbond. I'd expect the same powders to work for these bullets, with very similar charges.