Most all of my hunting was with a handgun until I was about fifty and my eyesight did not lend well to handgun hunting. I never liked a scope on a handgun (though I tried a few), mainly because it was just to bulky to carry and to nose heavy to sight without a support every time. Some folks do well with scoped handguns, but they are not for me. I have since turned more to rifles, but I learned a few things during those years of handgun hunting.
One of the things I learned was that, although a .357 could take down a deer, it was not the best handgun cartridge for the job. .41 mag is an improvement, but I settled on .44 Mag, .45 Colt and .454. A .429 to .451 diameter bullet is good medicine for most any critter in North America. I never used expanding bullets, but stuck with cast for .44 Mag and .45 Colt or jacketed for the .454, but never used hollow points for hunting; they are not necessary, the bullet diameter is sufficient.
I have handgun hunted Mule deer, Whitetail, Puma, Javelina, Black bear and elk with handguns, and I think the only critter out of those listed that might have been reasonable for a .357 would be Javelina. All the rest I would recommend a big bore.
Your 7.5” 44Mag should do just fine; practice with it until you get good. I would urge you to get good with what you have instead of buying another .357...of course you could buy it anyway, because, well because we always need another gun.