Win 73 ,when you used the 38/40 you probably shot the deer too far back or gut shot it. that could have happened with any caliber. would you say a 44/40 is too small? 38/40 and 44/40 have almost the same ballistics 38/40 -180 grain bullet at 1300, 44/40 200 grain bullet at 1300fps.
I used my '73 Winchester to shoot the deer. It was zeroed at 50 yards. The last three shot group had two holes overlapping and the third only a half inch from the first two. So I know the rifle was hitting where I aimed.
I know the .38-40 is a relatively weak round. So I waited for a good shot. I was in a built stand with a solid rest. Three does came out into the field I was watching. I waited until I got a standing broadside shot at 40 yards. I aimed right behind the deer's front shoulder. With that rifle (it is the rifle version, not the carbine, with a 24 inch octagonal barrel) at that distance I feel certain I can keep every shot in a two inch circle. So I know I didn't gut shoot it.
Some deer are just harder to put down than others. I have killed nine deer with my .30-06. Some of them dropped in their tracks, some ran. One ran about a hundred yards. The '06 has several times the velocity, energy, and shocking power of the .38-40 so I am not suprised that a deer could run that far after being hit by the .38-40.
I know that many deer and other game probably up to and including buffalo have been killed by .38-40 and .44-40. I am not saying they won't kill a deer. I was just using my experience with the .38-40 to show that the .32-20 with a smaller bullet and only half the powder would even less likely to cleanly kill a deer. Now there are just so many more better choices available that are much more likely to cleanly kill a deer.