I’ve been hunting deer with a pistol for over 25 years now. I started out with a 357 magnum Colt trooper and ended up settling on the 44 magnum T/C Contender. I also used a 41 magnum, 45 long colt and 357 maximum in between them. I am here to tell you all of them will take a deer as long as you take your time and place your shot well. Know your limitations as a pistol shooter and the distance you can shoot accurately.
I have used both hollow points and soft points over the years. Deer are a soft skinned animal. Believe me when I tell you both will kill a deer easily. In my opinion this all depends on where you plan on hitting the deer. Some people believe in shoulder shots to anchor the deer and have less possibility of having to track your deer. In this case you will need more penetration to get through the shoulder to the vitals. I would recommend a soft point for this type of hunting. Others believe in placing your shot just behind the shoulder. Access to all the vitals without having to pass through any heavy shoulder bone. In this case I use hollow points. specifically the XTP 240 grain bullet.
you‘re going to have to find a round that shoots well out of your pistol. The good part of that is that it will get you more practice time. If no one has mentioned that, you will need a lot of practice to hunt with a pistol. All guns shoot bullets differently. If I had to make a recommendation it would have to be the Winchester white box ammo. For me, it has always been the most consistent and accurate out of the box ammo I’ve shot. Give it a try and see if it will work for you.
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While my comment is not exactly in line with answering the original question in this thread (to use a ,357 Mag or a .41 mag or a .44 Mag for hunting), I do wish to point out that revolvers are NOT pistols. For those of you you have used the term 'pistol' always to maen a \handgun, this idea will be foreigbn to you and you will no doubt refute it.
But a pistol is a handgun that shoots the bullet from the barrel, as they did in the very olden days where every weapon was muzzle loaded, and there was no choice. Some dictionaries will say a pistol is a handgun in general, but this is not correct. And to prove my point here, ALL handgun manufacturers and all ammo manufactures never confuse these two, and revolvers are never within the category of pistols. They are both always in separate categories.
In fact, the ammo for these two types of handguns is almost exclusive, where revolver ammo is not shot in pistols and visa versa. Of course there are a few exceptions, such as the .45 ACP that can be shot in many .45 revolvers, the 9mm cartridge that can be shot in a SA revolvers which come with a 9mm cylinder, wand takes a cylinder swap, and the Desert Eagle pistols which are pistols made to shoot revolver ammo.
And to address the question placed in this thread, the .44 Mag has far more power than the .41 mag or the .357 Mag. I have a ballistics File I created, starting back in 2015, and it has over 3,500 entries, where each entry is a lonk to buy that ammo online, with its ballistics info include, and more. At any rate, the .357 Mag max's out at 907 ft. lbs. (PPU), the .41 mag at 1.150 ft. lbs. (Grizzly), and the .44 Mag at 1,533 ft. lbs. (buffalo Bore). So, from this you can see the relative power difference between all three calibers.
While I have not ever shot any deer with my .357 Ruger Blackhawk, I almost did once, but the cocking of the hammer scared off the deer (30 yards maybe). So, use a DA revolver, the Ruger Redhawk, which I believe comes in all three calibers. In addition, I have read elsewhere that hardcast (HC) bullets are the best for hunting big game!, and not to use hallow point. Of course, where you want to hit the animal may impact your bullet choice.
And my last point is about a comment in here about using the 10mm pistol for hunting, which to me is kind of a joke, AND probably just a biased comment since those that recommend it probably have one (and hence, they are biased), and most of them have probably never taken down big game with it. But at any rate, the max power of the 10mm is 728 ft. lbs. (Buffalo Bore and Grizzly), which is a far cry from the .357 Mag max. and of course from the other two even more powerful calibers.
My ballistics file is available to anyone who messages me and requests it. It is a PDF file, and it is free to all. I use it to buy the lowest cost ammo for my revolvers, and also when I want power or specialized bullets., or a certain brand, or... The bullet type is listed in each entry also. The file covers 35 handgun calibers, and 25 rifle ones also.