Shooter 2675, I have been around elk and elk hunting for over 50 years.
I can't even say how many I have killed, and I am betting I have seen about 5X more killed then I have killed myself.
I can assure you that either your 8X57 or your 375H&H are both all the gun you'd need to hunt elk. I have killed one good bull with my 8X57 and quite a few with my 375H&H, so I speak from experience here.
However, I also believe any reason to buy a new gun is good. So is "no reason"
The longest shot I ever made in my life at an elk was just a bit over 400 yards. Could I have fired at longer range? Sure. But why would I want to? Elk at ranges longer then that are easy to get closer to if you have 3 ounces of hunting skill.
I am quite a good long range shooter. I shoot to 1000 yards and even farther pretty often, but I NEVER fire at game over 500 meters away. There is no valid reason. In fact I would estimate that 95% of all the game I kill is at 300 and less ,and 75% is at 150 and less. Last year I didn't get an elk, but I killed all my game (deer and antelope) with iron sighted rifles and it was not even a problem.
When I advise new elk hunters about rifles I tell them all what I'll tell you now.
Any rifle that uses a bullet you can depend on to exit the elk after it expands is a good elk rifle.
Some of that will depend on the range of the shot. Expanding bullets penetrate LESS when they impact at very high speeds due to a larger mushroom and/or loss of weight. Impacts at between 1800 and 2600 FPS seem to penetrate deepest, and impacts at less then 1800 fps will go somewhat shallower. This is not "written in stone" and there are exceptions, but as an average you'll see this is true of most expanding rifle bullets.
A bigger diameter hole causes more blood loss and damages more of the organs. BUT elk are killed every season with nothing more then sharp sticks (arrows) and they die.
I have killed elk with 44 magnum handguns, flintlock rifles and a wood arrow, and I have also killed them with a 270 Short mag, M1 Garand and scope sighted 270s ---- and about 15 or 20 other rifles.
As a class the shells that I have seen with the highest degree of success in elk hunting have been 270 Winchester, 7MMs from the 7X57, 280s and 7Mags, the 308, the 30-06 and the 300 mags. Just as successful have been the 338-06 35 Whelen and the 338 mags.
Shoot a rifle that you shoot very well and by that I also mean using a rifle you can hit a 1 foot target with at 100 yards and do it in under 3 second.
One thing that is a matter of fact (for which I have no logical technical explanation) is that in all the elk I have killed I have not seen ANY discernible difference in how fast they fall after the hit with rifles from the 270 Winchester to the 338 Win mag. All physics and science should favor the larger more powerful rifles (and in the case of the elk I killed with my 375H&H it does.) But from the 270 Winchester to the 338 Mag I cant see ANY difference and I do mean none at all!
I have seen quite a few unimpressive kills with 7MM Mags, 300 Mags and 338 mags but all I would blame on the bullets used, not the gun or the cartridge itself.
A poor bullet fired perfectly from a 338 Mag is not as good overall as a really good bullet fired from a 25-06. Bullets are for making bullet holes. The bullet HOLE is what kills. Not the bullet. Not the scope. Not the gun.
Get a reasonably good sized hole (I would say at least 3/4" in diameter) clear through and out the skin on the other side, and it matters very little how the hole got there, Arrow, spear, handgun, rifle, shotgun slug........ All do fine if you can get them to go through in a fairly straight line without breaking up and/or turning off.
Bullets that turn inside the game are rare when they ball up and stay together but it can happen to them too. Just not very often. Those that break up often turn and go astray inside the game.
If you want to get a new gun (or 2 or 3 or 10) do it. Do it because it's fun and you should enjoy your sport and fun.
But your ability to shoot well (without the use of a bench rest) is FAR more important for success in elk hunting then the gun you choose.
The bullet is also far more important. Use a bullet that expands and still exits, and the rest is far less important. I killed elk with 44 magnum handguns with bullet that didn't expand at all, but 100% of then exited and all were killed with one shot each. Not one went over 25 yards after I fired either.
What is important to a point of being vital is your ability to use the rifle in your hand well. If recoil is not a bother for you ,then more power may be better. It never is bad for killing things. It can be bad for those that can't use it well. If you own a 375, and you shoot it well, I doubt anything is really going to hinder your marksmanship much because of recoil.
My daughter and both grandsons have killed elk, all with one shot per elk, with a 257 Roberts I made for her 30 years ago. Never lost one. Never had to shoot twice. All were killed with Nosler partition 120 grain bullets and Barnes X 115 grain bullets. So far none of them have gone over about 35 yards after they were hit with that Mauser and ammo.
As just one example, 4 years ago I helped a man (Hank) track down a 3 point rag-horn he shot with his 338 Lapua with a Burger. Hit was about 425 yards away from the hunter, and he fired 3 times, but only the 1st bullet hit. That elk went about 450 yards after the hit and we found it dead at the bottom of a gulch. Tracking was slow because for the first 75-100 yards of so, there was almost no blood. After that the blood started to come out the nose and mouth so the trail got a bit easier to follow.
So does that mean the 257 Roberts is a better elk rifle then a 338 Lapua? No.
But it is one of many times I have seen a lot of power not work as the hunter wanted it to. His shot was good with the bullet hitting the upper leg bone as the elk was slightly quartering toward him. But the bullet broke up and destroyed one lung but left the other one semi-functional.
Not the fault of the gun, the shell or even the placement. But the bullet hole was not what he wanted. Sure, he got his elk, but for the money invested and all the power of the round, and the $2800 he had spent on the scope alone, he was not impressed. I am not either.
The year before that, I killed my 2 elk (cow and a bull) with a 270Short Mag Benelli R1 Auto and a 308 bolt action Mossberg with an 18" barrel. The 270 had a 2X-7X scope set at 4X and was shot at a bit over 400 yards (the long shot I told you of above) The 308 was used to kill the other one at about 300 yards. That 308 has only a Weaver K4 on it. But both elk were killed, and neither went far after the hit. Both moved, but neither went more then about 30 yards. Hank was also on that hunt and saw it all. He brought it up too, after he saw his elk go nearly 1/4 mile the following season.
Both my 270 and my 308 were far less powerful then Hank's 338 Lapua but that didn't matter at all.
Don't overthink this.
Get a new gun because you want to. Don't get the idea you need to.
You have 2 elk rifles now. Bring the one you shoot best (new one or old one) from field positions, especially from standing, sitting and kneeling.