I've killed quite a few deer with the 357 magnum in both a rifle, and a Ruger Blackhawk. I've tried several bullet styles, and weights, at various ranges too.
Nearly any 158g JSP factory load will give you the results you're wanting on deer. I'd suggest the Remington load, because I've been using the Remington 125g, and 158g JSPs in various loads for a good while, and they're a lot tougher than most of the others(exept hard cast).
The Remington 158g JSP factory load should not only exit, but it should bust shoulders, and exit at reasonable ranges from either a rifle or a revolver. I'm loading this bullet to 1700 fps in my 1894C, and to around 1350 fps in my Blackhawk, and the performance at those levels is great on deer.
158g HPs are not gonna exit very often on a deer. I have had a couple 158g XTP-HPs exit, but that's out of my blackhawk, at revolver velocities.