I killed a 145 lb whitetail last year at 65 yards with a .44 mag using Rem 240gr hollow point bullets in a reload using a moderately heavy load of H110 powder. Hit was right behind the shoulder and the bullet was caught by the offside skin right behind the shoulder.
Expansion was perfect-it looked like an ad in a gun magazine. The expanded bullet was nearly 1" across in every directioon.
Deer ran (with both lungs penetrated) about 80-100 yards with a good blood trail. His path could have been followed by footprints in the leaves as he was stumbling and staggering, tearing up the leaves as he ran. Recovery was easy with the trail he left.
I might have wished for complete pass through for a better blood trail, but that was not needed. I consider it to be a near perfect outcome.
If you need/want better penetration, (and/or hunt bigger, heavier bodied game) use a more strongly built bullet or a heavier bullet. If you want faster knock downs, hit something solid such as centerpunch a shoulder or some such.
I think a 240 grain .44 is a near perfect combo for deer sized game. Good luck. Handgun hunting is a total blast! Have fun!