A couple of observations, not for hunting but firing magnums in general. Take them for what they are worth.
A cheap diagnostic technique can tell you if your trigger stroke needs some work. Dryfire with a dime on the flat top of the cylinder strap (just in front of the sights generally). If the dime falls off during the trigger stroke, then you need some work especially if you fire Single Action.
If you have a flinch, two things can help--first get some practice firing powder puff wadcutter loads (either in .357 or .38 SPC) and put a dummy round in the middle of them. Spin the cylinder so you don't know which round is coming up. Do this for awhile before stepping up to a heavier load.
Then, after licking your flinch. Step up to the .357 full bore magnum load that you want to fire and practice with it. Wear padded shooting gloves so you don't tire and hurt as much. Back in the day, the Newhall incident caused police to change training methods where they used to use 10:1 ratio of .38 spc to .357 loads fired during training. After Newhall, the std. protocol was to use std. issued ammo for training. I suspect that you will need to do the same if you want consistency in the hunting field.
Last, and this is simply my opinion, the K-frame is really not optimal for heavy .357 loads for hunting. It is a bit light for that--there is a reason that Redhawks, Blackhawks, Smith's L-frames and N-Frames, along with Dan Wessons, or even the GP-100 are built more for hunting and consistently firing heavy loads. Weight is your friend when firing heavy loads and many of these are also tapped for scopes or red dot sights. Longer barrels also give better sight lines and if an underlug design help fight muzzle rise. (BTW, I am not slighting Colts here--a Python is a marvelous handgun and the Trooper not far behind but these are becoming collectibles that running around in the hunting woods might damage the value of especially the Python).
Personally, I love the K-Frame in .38 spc, for .357, not so much. Ironically, the best grips, even better than Hogue, for me in magnum revolvers are the Ribber grips of Taurus--even firing a full bore .44 mag in a Taurus Tracker without padded gloves did not hurt. Wish Taurus would spin off their grip division and make them available for other brands.