CONGRATS! The .44 Special has always been my favorite round. Easy to load for and a bit versatile if you reload. I like mild loads and it's just fun to shoot.
I've wanted a 4 inch 624 for a long time. In fact I just brought home the identical 624 this week and it's on my hip as I type this. We live in a rural area in the back-country so coyotes are always an issue with our 60-70 chickens here. In summer add to that rattlers (a vet bill for the dog is around $2k last I heard - so just can't have them around), so around the house here a snake load is always first up during that time of year.
200gr Speer Gold Dots are my favorite for when I leave home, otherwise it's 250gr Keith bullets around 800fps for pests and plinking at home here.