Been handgun hunting deer for a shade over 30 years now, and have used almost everything mentioned thus far, sans the .41 magnum. I have a trusted friend who uses the .41 though, and he has killed more deer than I have with the others. He says it is as good as the .44, and I believe him.
The .357 did not impress me. It will kill them allright, if you place it with surgical precision- which is much of handgun hunting anyway. What it does not do is leave copious blood trails that Stevie Wonder could follow.The .44 mag and hot-loaded ,45 Colt (read RUGER) will do this, especially with expanding bullets which are heavy enough for full penetration. Big-meplat cast bullets are almost as good in this regard, and they will usually shoot through a deer from any angle. You absolutely want the bullet to knock TWO big holes in the animal, coming and going, and you want the blood trail that you get from such a wound. No conventional handgun is a guaranteed, drop-em-in-their-hoofprints hunting tool. How can I say this, if I haven't used the .460, 500, etc?
Because none of them are a .30-06. I shot a 160 pound doe a few years ago with a real destructive 150 grain '06 load. I hit her unawares at 125 yards, blew the heart to shreds and made an exit hole in the hide the size of a silver dollar. Reaction? She sorta humped up, them walked off and jumped flat-footed over a 5-foot hogwire fence. She piled up dead on the other side, but she show no immediate reaction to having her heart and three ribs scattered across the landscape.
She was dead on her feet, but having never read a ballistics chart or internet discussion forum, she just didn't know it. Sometimes they just need convincing, and the .40+ guns simply accomplish this with greater authority and reliability.
Get as close as you can- placement really is critical. So is repeating the application until the desired results are obtained.