A.J.,
You can't go wrong with the standard 240-grain JHP if you have a 6" or greater barrel. Just about all will give excellent results. I prefer the Hornady line, but only because it has shot well in my particular guns. Your Smith & Wesson may prefer Winchester, Remington, etc. Whatever brands you try, stay away from jacketed softpoint (JSP) loads. They rarely expand in .44 Magnum handguns. If you're using a 4" .44 then you may want to consider the Hornady 200-grain XTP load. The lighter bullet will help compensate for velocity lost in the short(er) barrel.
THE most important thing isn't ammo selection, within reason, but rather to practice enough to give your deer a quick, humane end when hunting season starts. Practice close at first, never from a bench after you set the sights, and work your way farther and farther with each box of ammo. I believe that the farthest distance you can put six shots into a six inch circle is the maximum range for your skill level. Don't be discouraged if that range is closer than you expected it to be. My own limit with iron sights is about 60-65 yards while standing and I've been shooting Magnum revolvers for almost 20 years. Handgun hunting takes a lot of field practice and for most people it is a relatively close-range game. The Elmer Keiths of the world are few and far between.So practice, practice, practice, and the best of luck to you this Fall. Keep us posted!