I would agree that a .44 mag is a far superior round than the .357 for hunting. A large selection of factory fodder is available and a handloader can pick and choose loads and bullets that perform best in his particular firearm.
The .357 will kill deer very well if the shot is precise but the .44 gives a little more room for error, with it's increased power. Just practice, practice, and practice more fore handgun hunting,, alot more room for error trying to line the sights, control the trigger, etc. with a handgun when your adrenalin is going nuts when a nice buck appears. Good hunting and good luck.