Seeing in your profile your from Texas, a 6.5x284 with a 7 or 7.5 twist to shoot the 156 Berger would be my choice. A 156 @2950-3000 fps would definitely take down anything Texas has to offer, and with exceptional ballistics. If you travel to hunt, it would be sufficient for elk as well, that is essentially 30-06 performance, I killed quite a few elk with a 30-06 and 150's or 165's, the 156 would be similar energy but with better sectional density and much better bc and energy at farther ranges compared to the 30-06 in the same weight.
Now, if your primarily wanting a rifle for larger game, I would look at the 30 cal, the only speed bump is the bolt face. Unless you go with something like the 300 Sherman (a necked up and improved by fireforming 270, you do already have brass.....) it would be better suited to go with a magnum bolt face 30 cal, such as a 300 win mag, which is doable, but then mag space may be an issue, I'm not sure what you have for magazine OAL. However if you did go with a 300 Sherman, all you do is neck up your .270 brass to 338, use the Sherman die to neck back down to 30 cal to make a false shoulder, then load and shoot it, and you have brass that you can load and shoot 215 grain bullets at 2900-3000 fps depending on barrel length. The downside is no factory available ammo, and a little more effort to make brass, but it's not bad, just means more shooting!!! Also you will never run out of .270 brass and 30 cal bullets.
But if you really don't want to form brass, that is a no go. A 30-06 is about the best performance you will get in a 30 cal without opening the bolt face, if bolt face is opened for a magnum 30 cal that means 300 prc, 300 wm, 30 nosler, and maybe a couple others could be decent options, just consider mag length. A 6.5x284 would be the easiest caliber change to get stellar performance from a 6.5 that would have readily available components and loaded ammo if needed. No concern on mag length.
