What ammo are you using? I suspect nerves and/or a misplaced shot are to blame.
It was a standard factory loaded Winchester 180 grain flat nosed bullet. I know it wasn't nerves or a misplaced shot. I was not nervous. I have killed 13 deer over the past six seasons. Never experienced the slightest "buck fever". I had been watching three does for 20 to 30 minutes patiently waiting for a good shot. When one of the does gave me a standing broadside shot at 40 yards I took it. The stand has a solid rest. I used it and placed the front sight right behind its front shoulder. As I said before, at that distance with a solid rest I am sure that I could keep the bullet within a two inch circle.
And, yes, compared to any of the rounds typically used for deer today, the .38-40 is relatively weak. And even to other cartridges of the time it was relatively weak. For instance the .45-70 had nearly twice the powder charge. The toggle link action of the '73 Winchester is not a strong action. That is why Winchester brought out the '76 Winchester. It was still a toggle link action but it was larger and stronger so it could fire a more powerful cartridge.
(Luke 11:21 KJV) When a strong man armed keepeth his palace, his goods are in peace:
(Luke 22:36 KJV) Then said he unto them, But now, he that hath a purse, let him take it, and likewise his scrip: and he that hath no sword, let him sell his garment, and buy one.