A .300 magnum produces roughly the same velocity at 100 yards that a .30/06 produces at the muzzle, and for that "advantage" you are rewarded with nearly double the recoil, which will feel much worse in a light, plastic stocked rifle, that in all likelihood is too long for you. A .300 magnum does have some advantages over the .30/06 for the long range hunter, but if you are expecting something different than you get on the target with your .30/06 you might be disappointed. Your shoulder will hurt and your ears will ring, so your senses tell you that you are applying far more power to the problem, but its an illusion, and what advantages there might be with respect to trajectory are difficult to exploit due to the heavier recoil.
The 7 mag produces flatter trajectory with recoil that is similar to the .30/06. This level of recoil makes it easier to exploit the (slight) trajectory advantage of the 7 mag. Another option you might consider is the .280 Remington. This cartridge is nothing more than a .30/06 necked down to 7mm, but its rewards are lighter recoil, good accuracy, and a trajectory that rivals and in some cases matches that of the 7 magnum.
So much for trajectory; if power is what you're seeking though consider an increase in bullet diameter and weight. Recoil increases with bullet weight of course, but if you can handle a .300 magnum, a .338, a 9.3 or a .375 is little worse. When loaded with suitable bullets, a medium provides an exploitable advantage on game, without the gore frequently associated with a high velocity small bore. If you prefer synthetic furniture, the Ruger Alaskan in .375 has much to offer, and when loaded with a 260 gr Accubond or a good quality 270 gr bullet, it shoots flat, and produces the ballistics that has accounted for all manner of game since 1912, when the .375 H&H debuted. IMHO, that rifle topped with a suitable scope is an excellent choice for big game of any size.