That's a nice little horn! I would agree with those who suggested it is a day horn or bag horn, meant to be carried in a pocket or pouch.
@oldwood , I would really like to see that old pen and ink drawing of yours, that shows the horn in the pocket! Another fellow on this forum recently showed a Tennessee horn of similar size, apparently made from an eastern bison horn. I think some of these may be mis-identified as priming horns for flintlocks.
Regarding the tip plug, people now appear to prefer a violin tuning peg, which is usually made of ebony. This is a
very hard wood. Its dimensional stability is one thing to make it desirable, I suppose, but the shaft is tapered and unless the hole in the horn is tapered to match, it isn't going to fit very well. You could shave or sand the taper off the peg, or you could invest in a violin peg-hole reamer (they aren't cheap) to taper the hole in the horn to improve the fit. The degree of taper is standardized, and reamers are available from suppliers to the luthier trade.
At the other end of the spectrum are the pegs crudely whittled out of a twig, as shown by Spence10, above. These work. However, I believe the typical practice in the old days was to carve a nice tip plug of
soft wood. For one thing, soft woods are easy to whittle, but another advantage is that the wood compresses easily, which facilitates getting a snug, airtight fit. An empty horn should make a
PUNG sound when the stopper is removed. My dad grew up shooting muzzleloaders, and told me many years ago to use softwood for this purpose, and he insisted on an airtight fit for both base and tip plugs. Ned Roberts also noted the use of softwood tip plugs, in
The Muzzle-Loading Cap Lock Rifle (page 51, in my 1952 edition): "...the point of the horn cut off leaving a small hole through which it was filled and the charge poured out,
a soft-wood plug whittled to serve as a stopper for it..." Most of the old ones I've seen also had some provision for attaching a thong or string to the stopper, to prevent loss.
In any event, your new horn really is a gem! thanks for showing it!
Notchy Bob