One thought is the .357 mag isn't what it used to be in terms of loadings. In its early years the pressure limit was 46,000 cup, and now it's been throttled back to around 35,000 psi, in deference to small framed revolvers.
If the OP does not handload, then I'd suggest he round up some DoubleTap or Buffalo Bore ammo, to get the hottest stuff available. His 686 is decently strong, and should tolerate them at least occasionally. On small southern deer, I'd stick to reliably expanding hollow points, over hardcast.
If I was determined to take a deer with a .357 Revolver it would be either a six shot N frame or a revo of similar strength. And I'd load it as heavy as I could get away with loading.
