The .44 will do great.I have also hunted quite a bit with a 4" S+W Mountain gun,mostly using 240 gr soft points or hollow points to take antelope,mule deer,whitetails and bear.
Even with soft points,the majority of hits blew out the other side.These shots were from 30 to 75 yds.
There was one mule deer buck that I liver shot and when following it up,made a desperation shot at 150 yds as it was going over the next hill,and dropped it.
That was when my eyes were a lot better and I would never have taken a shot like that but the deer was already hit.
These days,I will shoot out to 100 yds if everything is perfect-a very steady rest,a calm,broadside animal and I have to feel "on".When I set up to watch a trail,I like to setup 40-50 yds.
The 4" .44 Mags are definitely up to the task.Just use a good bullet,and shoot within a range you are confident in.The gun is up to the task.