I'm transposing 44 advice for 357 advice here, but I have a bit of handgun-for-deer experience and I'd also recommend JSP's. My kills with them in the 44 mag revolvers I own have been definitive.
Also, if you're thinking HP, consider the XTP, which is made for more penetration while retaining some of the positive characteristics of a traditional HP. Had good luck on two deer with that as well.
I also have a Blackhawk 357, and intended to hunt with it this year, but instead stuck with the 44 mags I own, just because I'm very familiar with their performance. I loaded up some stout JSPs and XTPs for the 357 but just couldn't bring myself to do it. That said, the 357s JSPs worked great on a red fox at 35 yards....but a fox ain't a deer.
Maybe next year, and only at 50 yards or less.