I have a similar rifle, built for me by Curt Hardcastle of Silver City, NM.  Curt isn't fast, but his work is excellent quality and I'd recommend him to anyone.

Any good single shot smith can re-barrel, but I ran across an issue with one I have--the taper on the takedown pin isn't standard, so Curt and I decided not to maintain the takedown feature.  He welded that hole closed and build a very nice rifle, fitting a new Shilen barrel that was contoured half octagon by CPA.  I restocked it with nice straight grained English walnut from Treebone Carving.  With trigger work, it turns in groups of 3/8" at 50 yds. with ammo it likes.

A good smith can also adapt it to centerfire.  I'm not a fan of larger calibers in The small Martini, although the action is plenty strong.  Had you not said you already have a K Hornet, I would have recommended either the standard Hornet or the K Hornet.  You could rebarrel to anything in the .222 family.  The .222 Super (rimmed) is easiest, although you can have a smith build an extractor for the rimless case.  Pictures of these extractors can be found.  It just costs money. 

Clarence

Have you tried to do a thorough cleaning of the existing bore? Unless it was last used with black powder or corrosive primer ammo I question that siting for a long time would hurt it unless it was in a damp area.

Hi , just posted on another thread ,but feel the same info might be useful here .. the 32-20 is an easy conversion but use a 30 cal barrel ,much better and available projectile choices, I use cast bullets 90 gr for small game and jacketed 100 and 110 for small deer .. The 6mm 223 is also very good but needs a good smith to do the extractor job .. Rob

I would recommend (You need to Login or Register to view media files and links) for the rebore, they do great work.