Sheesh, that's a long, long straight-walled cartridge. Kinda like the John Holmes of the cartridge world ...
Cartridges of the World lists it. Not a lot of info.
Introduced in 1886 for the Winchester Model 1885 single-shot (designed by John Browning, don'tcha know). Discontinued in that chambering in 1904. Never very popular.
COTW says it's bottlenecked, but the old catalog illustration doesn't show that.
Old Ideal catalogs list bullet 375248 as proper for this caliber, so from that I'd guess it takes a bullet of .375 to .378 inch diameter, just like the .38-55. The old .38-55 was all over the map for bore size, and I suspect your rifle wasn't held to exacting standards, either.
So, first course of action is to tap an oversized lead ball or slug (never copper-jacketed) through the bore, and measure the slug when it emerges.
Standard bullet weight was 217 to 218 grains.
COTW notes the use of Lyman 37584 bullet with 21 grs. of IMR 4198 for about 1,350 fps.
With smokeless powder, you'll want to use a bullet that is slightly hard, with a modern, petroleum-based lubricant on it.
The old Alox/Beeswax-based lubricant works well with smokeless powder. Black powder lubricants may not work as well.
You have to tailor the bullet hardness and lubricant to the propellant, or you'll likely get bad leading. A hard bullet, with Alox/Beeswax lubricant, over a load of black powder almost always leaves a bore streaked with lead and the target relatively untouched.
The factory load undoubtedly used black powder. I'd suggest FFG grade. With black powder, the bullet should be very soft, if not pure lead. Lubricant should be SPG, Lyman Black Gold or some other non-petroleum lube that is soft and will keep black powder fouling soft.
Might want to use a Magnum Rifle primer to ensure all that black powder gets ignited well.
From its designation, we can infer that 90 grains of black powder was its standard load. Might want to put a "kicker" of 10 grains of FFFG black powder in the case first, to help ignition, then add the remainder FFG.
A .36-caliber wool felt wad, soaked in black powder lubricant (SPG or Lyman) might be a good idea twixt bullet and powder. If contamination by the lubricant concerns you, put a dry wad in first, then the greased wad on top of it. This will create a barrier.
You'll have to sacrifice 5 grains of black powder or so, to get them to fit in the case, but it's worth the extra lubricant or peace of mind against contamination.
Or, if you punch your own wads (a 3/8" hole punch is the right size) just cut a thin disk of waxed cardboard from a milk carton. The waxed paper wad, under the greased felt wad, will keep lubricant from soaking into your powder. Were I you, I'd use a thin, waxed cardboard wad instead of a dry felt wad.
But really, I don't think it's a big problem. I regularly load my black powder .45-70 cases with black powder, with a felt wad lubricated with Gatofeo No. 1 lubricant (a homebrew lubricant named after me but not offered commercially) on top of the powder. Never had an ignition or burning issue by doing so; all the powder burns.
I've fired cartridges as old as five years, without any perceptible effect.
The lubricated felt wad between the lubricated bullet and powder, will help keep powder fouling soft. It's asking a lot of a bullet to carry enough lubricant to keep soft the fouling that 90 grains of black powder creates.
About half of black powder is left as fouling, so that's quite a bit of gunk down a .37-caliber bore.
I looked through my directory to books and magazines. Couldn't find one article or mention of the .38-90, as far as loading it went.
Sounds like you have a relatively rare rifle, or at last the caliber is.
Not sure where you'd get cases. Might try looking online at Bertram's, Buffalo Arms or Midway. They may have .38 Basic cases that could be trimmed to fit.
Reloading dies will be pricey. Of all the makers, CH4D probably will have the best deal. Not sure if you could use .38-55 dies to kinda neck-size the .38-90. I don't know how the dimensions compare.
Check Gad Custom Cartridges for possible store-bought lead bullets.
You'll find it in a search.
It's late. I'm tired. Not giving you the links because I know you can search and find them easily enough.
Good luck with that .38-90. Sounds like a fun project.