I've had 7mm Rem mag, 300WM, 300WSM, 30-06, 280 and several others. Sold 'em all except for one 30-06 with too much history to sell and use 308 and 6.5CM now for everything. The 6.5CM lives up to the hype, but my go-to rifles are all in 308. Not because I think the round is any better, I just like the rifles I have in 308 better.
Either of those 2, will kill deer or elk past 500 yards. As will many other rounds such as 7-08, 270, 7X57. It is really more personal preference than anything else. The long action cartridges like 30-06, 270, and 280 offer only marginal gains over similar short action rounds like 6.5CM, 260, 7-08 or 308.
Your 308 is cheaper to shoot, ammo will cost less and if you handload you'll be able to use less powder. I use 58-62 gr of powder for every 30-06 cartridge I load. 40-46 gr of powder for every 6.5CM or 308 round I load. I get less recoil and better accuracy. And in the real world no animal will know the difference between any of them.
If you're going to move up to something bigger one of the 26, 28, or 30 caliber magnums is really the next step. I'd skip right over the 30-06 class of cartridges. At under 500 yards no animal will notice. The magnums only offer an advantage at 600+ yards.
Once again it is personal preference. The 7 mag is probably the smallest step up in performance, but the most commonly available. Recoil isn't that bad. Virtually the same as 30-06. As is performance. The 30-06 and 7 mag shoot the same bullet weights to the same speed. The only advantage 7 mag has over 30-06 is better aerodynamics of bullets in the same weight. I'd go to one of the 300's if I were moving up over 308.
But...most of the 30 caliber magnums will give roughly double the recoil of your 308 depending on the exact load. And use almost twice as much powder if you handload. Be honest with yourself. The rifle you have is good for at least 500 yards on anything you'll hunt, and will work for targets to around 1000 yards. Can you really shoot well enough to hit game farther away. And how many opportunities will you have to do it.
Consider what is actually on stores shelves right now. The rifle does you no good if you can't find ammo or components. Or if you can't afford them. Some of those cartridges are selling for $100-$150 for a box of 20 rounds.
That's what I did when I settled on 308. I have several rifles set up for different purposes. One has a short 18" barrel and a low powered scope for woods hunting. One is a heavy barreled rifle with a more powerful scope for long range use and a couple of others somewhere in the middle for general purpose hunting.
