My personal choice would be the .38-55 as it can be very accurate, once you've worked up loads for it. Some of the calibers offered can be a bit wimpy, and others can get expensive to buy brass for. Bullet selection, and molds for the .38-55 are pentiful, and it isn't abusive recoil either.
Of course this means buying the Hi Wall version, which I would reccommend anyway. It's more desireable, and stronger. You'll never wear this gun out in .38-55 caliber! Too bad they don't offer it with a hard rubber buttplate!

I vote with Sav99, first choice is 30-40, seond 30-30. I have two original 1885 30-40s, one custom, one factory, plus a bunch of Krags. Very versatile cartridge, you can load across a wide range of bullet weights and velocities, from .32 revolver power up to, and maybe a bit past, a .308 WCF. Immense number of cast bullet designs. Over a century of load development and no messy black powder. Sav99s second choice, 30-30, is just as versatile at the lower end but if you ever get the chance to go North to Scandinavia for moose ("elg" up there) it will fall short in energy. Both cartridges are equally extremely accurate in a HiWall. IMO a shotgun style butt is a necessity, a windage and elevation tang peep sight very desirable. Marble's makes a pretty good one. Avoid a vernier sight, they are no good for a hunting rifle. The Uberti HiWalls are not identical mechanically to the original Winchesters, parts do not interchange, but they have a generally good reputation.

Hello!

I have a Uberti 40-65 Highwall Special Sporting intended for some serious Longrange target work...

I selected the 40-65 instead of the 38-55 because both had 18" twist and the 40 then had the possibility to stabilize a heavier bullet, if the 38 had a faster twist I would had choosen the 38-55.


200m target I shoot in a Competition in france, the 10-ring is 80mm


Some bullets tried in my rifle
From left:
RCBS 40-350-CSA - 370gn
JKA 40-411 Mountain Molds, my own design.
RCBS 40-400-CSA - 420gn
RCBS 40-400-BPS - 417gn
Lyman 410655 - 423gn

Express,

A local (CT) gunshop has a Uberti High Wall for sale in 38-55. It's an "American" model and is marked Uberti Lakeville, CT.

My point is that there is way too much drop at the heel and no cheek piece. When looking thru the tang peep there is no way I can even get my face close to the stock and therefore my eye wanders in the peep.

I would handle one before buying it as this is a real problem. Perhaps the ones you have there are not like this used one. They are asking $850 and it's not new.

The rifle butt plate has pin that sticks out into your shoulder. I think it's lever for a trap butt plate.