Smoke and Recoil,
A .263 came out at .257 with no rifling marks? The 5-digit serial number says it was
made between 1892 and 1895. The rifling may have been corroded badly at some point, as it is certain to have been fired with corrosive priming back the day. I would be inclined to run a borescope through it and see what you can.
If it is worthwhile to you, you can get it relined and keep the chambering for less than you can get it rebored and cut rifled to a larger chambering.
This outfit makes the liners that Brownells sells and will do the drilling and installation and rechambering for $375. Their 25-20 liners are patterned after the original Winchester rifling, so it will be a restoration of the original configuration, except for being lined, of course.
If you have a lathe, you can get the parts and do it yourself, but it may not be worth it. The liner is $125 from Brownells, but if you don't already own an appropriate piloted drill and don't have a 25-20 chamber reamer, and a barrel vice with an insert for '92 barrel, it's less expensive to pay people familiar with the job to do it.
If you decide to go ahead with it, one thing you can do in advance of the work is apply penetrating oil to the barrel at the receiver shoulder and inside to get it ready for removal. There is a YouTube video that showed good old Liquid Wrench actually outperformed Kroil and all the fancy and expensive penetrating oil sprays, so I would just get a can of it and apply it once a day for a week and let it soak in.