Most factory ammo for the .38-40 is loaded pretty mildly. I know winchester used to make a jacketed soft point for .38-40, not sure if it's current production, but Midway still lists it as of today. It should push around 1150fps (published), meaning on the order of 550ft.lbs., plenty to kill a deer.
Looking at the other loads available at midway (all low recoiling "cowboy loads"), the Winchester Super X looks to have the highest velocity/energy. The others are in the 800-900fps range with 180grn lead bullets, to the winchester's 180grn jacketed soft point at 1150.
Yes, a .357mag DOES have more power, but the .38-40 worked well for deer for a LONG time before the .357mag ever came around. You might consider contacting a local reloader to press you some custom ammo, or see if Hornady offers the .38-40 for custom loads. My Speer manual says it can get 1514fps from a 180grn lead FP from a 24" barrel. That'd push you up to the 900ft.lbs. range, and give you a lot more range with the historic old rifle.