It is really not possible to place the modern, so-called Hawken or plains rifle in any historic category. As tg pointed out, the stocks are completely incorrect. It would seem that these rifle stocks were laid out with the main concern being getting the largest number of stocks out of a plank Or efficient use of scope sights. For the most part, the locks are all wrong, with Italianate "engraving" cast into the plate. It is hard to find any correct details on these guns.
They make useful, reliable and accurate field guns, but won't pass muster at a juried re-enactment. For a good idea of what the rifle should look like, try Don Stith's web-site, St. Louis Plains Rifle Company. Or the Hawken Shop. Typing these names into your browser window should get you there.
I'm currently refurbishing a T/C Seneca. It is a light, handy, accurate rifle--supposed to be a copy of a New England hunting rifle of the late percussion era. It's not. It is a generic, half stock interpretation from that era. Nothing I can do will change that fact or disguise the rifle's origins. But it will still be a great little gun for hunting, plinking and general woods bumming. It's a good gun, but it isn't authentic. I will still enjoy it every chance I get.
The problem for most of us is the tremendous price difference between an authentic rifle and the generic ones. And the cost is merited. In any school of rifle building, there are many, often tiny, details and nuances that take awesome skill to produce and without which the piece is incomplete and incorrect.
I think if it were me, I would enjoy the rifle I have and start stashing extra cash away towards a more authentic piece. And, if I already had the money, I would start watching the blackpowder sections of the on-line auctions or Track's web-site for appropriate guns. Thay are out there and sometimes you can even get a deal.
I truly hope this isn't too harsh an appraisal. I clearly recall how proud I was of my new T/C Hawken thirty or so years ago. And how surprised I was when people rather snidely commented on its lack of authenticity. I knew it wasn't an accurate replica, but the sheer nastiness of the verbiage took me aback. I hope I have avoided that here and I hope this is of help to you.
If you go to Track of the Wolf's web-site and look under the Guns heading, there are some good photographs of accurate replicas there.