the simplest (and generally best as per KISS principal) answer is a 44 mag, or 357 mag rifle. Ruger, Marlin, Henry, and others make great bolt guns or lever guns in those calibers. Or a good ol' shotgun with a slug.
pistol and black powder options are different, of course. i.e. if you get a 308 pistol, it's fine for deer, but not in a rifle
there's plenty of other options as well, 458 SOCOM, 358 Hoosier, etc. but those are less commonly available and have limited uses in other applications.
Seems like a Handi with a paper patched soft lead bullet up into the rifling rather than taking up case space would have lots of potential. I should try that.
I know of a few deer that would vouch for the effectiveness of my 358 Hoosier, but dead deer tell no tales. There's quite a bit of info about the Hoosier in various threads on this board. Basically a 358 Win shortened to 1.800 case length.
FWIW you can run a .35 Remington shortened to legal length
That allows for some pump, classic auto, lever and single shot rigs.
Reloading deal............but there are places that will trim and reload such ammo for you .
There are probably dozens of legal deer catridges, which have been discussed at length. If you're up for some light reading take a look in this forum and in the great oudoors forum.
The basic questions are: what type of rifle do you want to shoot (single shot, lever, bolt, semi, AR platform), how far do you want to shoot it (in yards), how much do you want to spend, and do you reload? Otherwise we are grasping at straws for a recommendation.
.450 Bushmaster will be near the top end in price, and middle/upper of the road for range depending on the bullet. As a bonus, if you reload you can use the same bullets in sabots for ML season.
If you're going for least expensive, single shots in .357 Mag or .44 Mag, or use a ML during firearms season. Next up in price is a .357, .44, or .45 Colt lever action or bolt action (Ruger 77/44 and 77/357).
.454 Casull and .460 Mag can get you out to 200 yards, or more with the right bullet in the .460... but pretty stout recoil.
An AR platform will probably be the most expensive (.450 BM, .458 SOCOM) and will get you 200yds plus in range.
A .357 Max, .357 Herrett, .35 Rem trimmed to 1.8" will go 200 yards and can be done with relatively little expense.
Stepping up in performance, the .358 Hoosier (close to .358 Winchester performance), .358 WSSM (duplicates the .358 Winchester or a bit more), and .358 WSM 1.8" in that order (the .358 WSM 1.8" can outperform the .35 Whelen). The WSM is a legitimate 50 state rifle on any game in NA. As is the WSSM at shorter ranges.
Anything legal under the current regulations is an effective deer round if the shooter puts it in the vitals and uses the round at an appropriate range for that cartridge.
I love my .450 Beast master (Bushmaster) upper on my AR. It is absolutely devastating on deer. Just dandy when you don't like to follow blood trails or you are hunting smaller wood lots and you don't want wounded game crossing over onto the neighboring property. I have shot three Whitetails and a Mule deer with it and it has performed flawlessly. Effective range with a good quality scope is in the 250-275 yard range; depending on the shooters level of experience. Some claim 300 yards but those big 250 grain bullets have a rainbow trajectory at the longer ranges.
Recoil is manageable, ammo is affordable and uppers will set on a standard 5.56 lower with 0 mods.
Cool factor is high too. Nothing like hunting deer with an AR platform.