I hope that when you change the scope you also change the rings. My objection to variable scopes with large objectives is that the require the use of medium to high rings to clear the barrel. This plays hell with the abilty to snap shoot with the rifle.
A heavy rifle should have the scope mounted so that the shooter's head position is as close to his iron sight head position as possible. Practice with the rifle at the International Skeet "ready position", with the stock touching the point of your hip bone. At someone else's signal, mount the gun to your shoulder. If all is well, you should be looking straight down the center of the scope, with your face firmly down on the comb of the stock.
One of the handicaps of using a higher power scope is (1) shorter eye relief and (2) smaller exit pupil. The exit pupil is the little circle of light you can see on the eyepiece of your scope. Your eye has to align with this little circle for you to be able to see through the scope. The size of the exit pupil is determined by the size of the objective and the power setting of the scope.
If you are using a scope like the Leupold 2 1/2-8X32 VariX III, with the power set on "8", then the exit pupil is 32/8=4mm in diameter. If the power is set on 2 1/2X, then the exit pupil is 32/2.5=12.5mm, actually larger than the pupil of your eye, and very easy to pick up, which can be important in a tight spot.
Benchrest and target shooters use up to 32X scopes with 50-55mm objective lenses, which results in tiny exit pupils, but this is not a negative factor, since they have all the time in the world to line up with the scope.