Like with a little bit more velocity and less bullet drop past a certain distance.
Unless you are willing to go up to a magnum round nothing will significanlty better the 30-06. A 270 or 280 offers VERY slight advantages in trajectory, but with lighter bullets and less performance at long range, and on larger game.
My first thought is .308. Better ballistics, shorter action, fairly common (plus milsurp). It'll be my next deer rifle.
Sorry, but this is a step down. The 308 is a great round and one of my favorites, but it shoots the exact bullets as a 30-06 but about 150 fps SLOWER, not faster.
Same for 338-06 and 35 Whelen. They shoot heavier bullets, but offer zeo advantages over 30-06 when it is loaded with the heavier .30 bullets. They offer very poor long range performance. And while they don't have magnum stamped on them, recoil is comparable to a 300 win mag.
The next logical step up over 30-06 is one of the 7mm magnums. They shoot flatter than 30-06, 270, 280, or 300 magnum and recoil is so close to 30-06, you'll probably never notice the difference. The 300's heavier bullets offer a small advantage in energy numbers out to about 400 yards, but beyond that the 7 mags not only shoot flatter, but hit harder. The 300's are also a step up in recoil.
I've owned them all. Personally I'd keep the 30-06. If you don't handload, take the money you would have spent on another rifle and start loading for the 30-06. It is easy to safely beat factory speeds by at least 100 fps with 150-180 gr bullets and good handloads. You won't need anything else.