I’ve successfully shot bull elephants with a .375 Holland & Holland Magnum.
I formerly preferred Remington 300Gr round nosed steel jacketed FMJ solids (which may have been loaded with Hornady bullets but I have no way of verifying that).
Now, I prefer Norma 300Gr Monolithic Solids (among factory loads).
.375 caliber is perfectly adequate for brain shots on even the biggest bull elephants from all angles. But it’s very marginal for body shots on elephant (I speak from personal experience). So I can’t recommend it as a serious elephant hunting caliber.
For elephant, my preferred caliber would be either .600 Nitro Express (in a double rifle) or .505 Gibbs (in a magazine rifle and only with 600Gr bullets). Even though I can comfortably handle the recoil from both these calibers, I must let sanity prevail and it would be criminally irresponsible of me to recommend them to someone unless they are fully comfortable with the amount of recoil that these two calibers generate.
So, for most hunters I’d recommend the .458 Lott as a good all-round elephant hunting caliber if you prefer magazine rifles. If you prefer double rifles, then .500 Nitro Express.
Your .458 Winchester Magnum is perfectly adequate, though.
As for bullets, if you’re limited to factory loads… I really like Norma monolithic solids. If you hand load, then I strongly recommend the Cutting Edge Bullets Safari Solids (but just don’t go over 450Gr in the .458 Winchester Magnum)
Double rifle or bolt action is a matter of individual preference. But use the platform which you’re most comfortable with shooting quickly and accurately. If it’s a double rifle, then just make sure that it features a non-automatic safety.