.22 Hornet in Ruger #1 SS
.25-20 in Savage 23B, Win '92 rifle, and Win 1885 SS rifle.
.25-35 in Greener Martini SS
.38-40 in Win 1885 SS
.44-40 in Win 1885 SS and Colt New Service

I have kiled deer (mostly CA blacktails and NY/VT whitetails in "jungly" woods) with all of these in the past, but only use the .25-35 and .44-40 now. Both work very well if you handload and have a strong-actioned rifle.

My .44-40 1885 Hiwall is easy to get to .44 Mag rifle ballistics (but cases don't last as long as the Mags would). But I just load it to the same ballistics as the old .44-40 WHV loads meant for Win '92s, using Hornaday 180 gr. jacketed hollow points. At .44-40 velocities they expand perfectly (my Hiwall is a reproduction with .429 barrel, so .44 Mag bullets work fine if you can make the crimping cannelure fit).

In a single shot you can use pointed .257 95-100gr. bullets and a somewhat heavier charge in the .25-35 and get much better ballistics than the underloaded factory 117gr. flatnoses.

I've never used a cartridge more "modern" than the .250-3000 (1913) for deer hunting. Biggest deer I ever killed was with a 16 guage LC Smith SxS! And I've killed a couple with a Ruger Bisley .357. Works fine if you are close.