I will relate my personal experiences with the .30-06 at close range:
1. Forkhorn, 165 dressed. (They grow big, but with small racks in Central Minnesota.) 65 yards measured, quartering away shot. My handload with 165 gr. Nosler Partition, 46.0 gr. IMR3031, Federal brass, CCI Large Rifle primer.
Bullet entered just behind the right front leg, took the top of the heart off, exited the chest through a rib, went into (Would it be the femur or humerus on a deer?) bone of left leg, shattered it and lodged under the skin on outside of the left leg. Recovered bullet looked just like Nosler's ads. Deer dropped on the spot, did not take another step.
2. Spike buck, 185 dressed. Same handload. Straight right broadside shot, same aiming point. 75 yard shot. This deer, however, had just been chased through a swamp by a hunter, and was in high alert. (The first one was following a doe, with amorous intentions.) Bullet took off top of heart, deer took off like a bolt of lightning, ran 75 yards and hid under a log. Since it was dusk, I didn't find him until the next morning. Bullet had exited through a rib again, but behind the left leg, so was not recovered.
No more deer with that load, but I did shoot a coyote that came into my bear bait with it; 20 yards, chest shot, lungs were jellified, bullet not recovered. DRT.
I think using a to heavyweight bullet in the 06 would more likely cause you to have to track a whitetail or possibly loose the animal. The 150 gr. Softpoint is just about optimal in IMO for east woods whitetails.
I had a friend who used nothing but 220 gr. round nose in his Winchester 100 .308, and never had to track more than 100 yards. He swore by them, only load he used.