Assuming the handgun you listed IS YOUR CHOSEN WEAPON, then rather than throwing out a lot of suggestions for alternatives, perhaps the space would be better used discussing the question the OP actually asked?
So.
The snubby can be used as a game getter, it just takes practice to know its limitations. Inherently, it can be as accurate as a handgun with a longer barrel and better sights. You just have to pick a load that will do what you want, and shoot it enough to know you can hit with it.
I have hunted small game many times using the .38 Special cartridge with good results. I have always used wadcutters or semi-wadcutters. You don't need expansion, just accuracy, to take a rabbit or squirrel with a bullet that diameter. You also don't need a great deal of velocity, just a mild load.
For humans, the load you already have listed is quite adequate.
For larger game, you'd be lucky to ever get close enough to use this handgun, but... a heavier bullet will likely give better penetration. I would not use a hollowpoint, as penetration is what you want. So I would explore the accuracy of some of the loadings available in a jacketed or lead solidpoint. There are lots of bullet weights available, from 110 grains on up to 158 grains, generally. I would experiment with those loadings and see which is most accurate.
The handgun you have chosen is not perfect, but it is the one you have chosen, so you must learn to live within its limitations, or buy another handgun.
Andy