The TC stock design is modern. The TC people apparently did not bother to look at a real Hawken or any period guns. They apparently were not shooters either. The stock is uncomfortable and looks wrong. The scallop where your thumb goes is thoroughly modem and not seen on old rifles.
Somebody mentioned Enfield rifled muskets. Those stock are of extremely poor design. The idea was to use an open "tactical" shooting stance. IT is quite a pity. I have minty Parker Hale 2-band. I could not bring myself to modify the stock. So, I added sight height extensions.
To make the TC work the comb line needs to be dropped. She my doctored photo. The actual comb can be move back a smidge to make it look right. The comb needs to be thinned a lot, see the photo of the SMR for inspiration. The cheek piece is much to large. IT will also need to the thinned to blend at the front. Original cheek pieces are small and thin, mostly not functional. Original stocks were dainty by modern standards. While you are working on it, eliminate the flat spot along the barrel channel. That is also modern and inappropriate. Fix the lock panels, they should be very thin.
Don't worry about copying an original Hawken, the TC is not a Hawken. For inspiration though, here is link to a real Hawken.
https://auctions.morphyauctions.com...___HIGH_CONDITION_SAMUEL_HAWKE-LOT470518.aspx