Probably a fast-twist barrel. Me, I'd buy a box of 220-grain and see how it performs. Remington CoreLokt would likely do fine, although you might have to ask around more than just one gunshop.
Also might try 180-grain boat-tails, since they're a longer bullet.
You can check the twist rate with a cleaning rod. Make a marker with tape, and then ink a mark on the side of the rod so that you can tell when the rod makes one full revolution as you push it down the barrel. At one rev, wrap tape at the muzzle. Withdraw the rod and measure the length of insertion. That's the twist. One turn in xxx inches.
Common for the '06 is 1:10, which works for 110-grain through flat-base 180s for good groups. I suspect your rifle might be as fast as 1:8...
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