The handgun will do the job, but there is no margin for error.
This applies to all of them, regardless of caliber.
One thing I think is of some importance, is which .357 Magnum we are talking about when someone asks if it "enough" for deer.
Are we taking about the .357 magnum as it was back in the late 30s, when it was used to take virtually everything that walks in the Western hemisphere?
Or are we talking about the .357 toned down so you can shoot it in lightweight snubnose revolvers?
They share a name but little else.
My father carried a .357 while deer hunting for over 30 years, I never heard him question it, not even once. Nor did several very dead bucks. Of course, this was also decades before the Internet.....
Dad did switch to the .44Mag, (I think more out of curiosity than anything else), and after he had taken his second deer with it, he asked me if there was any .44 Mag load (factory, Dad didn't do handloads), that
wouldn't exit the deer. I told him probably not, but to try the 180gr JHP.
sadly, he never got the chance to find out, as his health worsened soon after and he never got another deer.
I believe that the .357 Magnum (in full form) is adequate for any deer the shooter is.
If you want another plus for the .357, if needed, it is considered a better choice for self defense than the big .44.
I've got .357s and .44s and love them all, and I find them both quite good for what I use them for.
I would recommend checking your state's game laws, as there may be restrictions on both caliber and gun. And, they can change.
One state I lived in wouldn't let you use anything with less than a 6" barrel. At the time, I wound up swapping off a Ruger .41 Blackhawk 4 5/8" for a .30-30, so I could hunt deer....TODAY I wouldn't have to do that, in the same state, they changed the game laws.
SO, besides the choice of caliber, choose a gun that complies with the game laws, too.