I've shot quite a number of bears, including browns, grizzlies, and blacks. The prior post covers a consideration I completely agree with. After a bear has been hit, it will generally continue to scramble until it reaches cover or dies. They don't typically offer high percentage follow-up shots on non-fatal hits. If they reach cover and haven't been hit pretty darn good, they don't leave a great blood trail, and they're tough animals that can motor long distances through the ugly stuff. Unlike deer, moose, caribou, sheep, goats, etc..., which will often stop and look around after being hit, bears head out for the safety of cover, and then continue to put distance between themselves and danger. Perhaps because their eyesight isn't the greatest, they don't stop to look the country over like other animals more dependant on their sight for survival.
So if you're confident you can place a fatal first shot hit on the bear at longer distances, I don't believe you'll have any trouble killing your bear with your '06 provided your bullet has sufficient remaining velocity to expand upon impact.
My primary advice is don't expect a non-fatally hit grizzly bear to pose and offer standing-still followup shots. Expect the bear to be on the go until cover is reached. Based on my experiences and observations, I recommend getting within sure-kill first shot range, so that if something does go wrong, your chances of hitting the bear with a followup shot on the run are improved.
But if you can't close the distance to the ideal range, I wouldn't hesitant to strike a well placed killing blow at longer distances with your '06. The average grizzly isn't a huge animal. If you bump into an old mature boar, then yes, they can be down right impressive. But not on the same order as a mature brown bear boar. The big coastal brown bears can be unsettling to the uninitiated simply by virtue of their immense size. A mature coastal brown bear will dwarf the largest of inland grizzly bears - no comparison. And your '06 will kill a big coastal brownie with a good hit, so don't feel handicapped with the '06 for grizzly bears. Place the shot properly and enjoy your hunt. It should be a great time.