I don't usually carry a handgun when I am hunting.
When I do, I usually carry a 4 in S&W 651 .22 WMR. Mostly so I can legally take grouse. I switch to a .357 Taurus TI Tracker/9mm/40 when packing meat when I don't have a long gun.
Based on your considerations I would recommend you consider a Ruger LCRX
3in .38 Special +P. 17 oz, enough power, versatile, etc.
