According to Mr Knibbs prefix S is .22 and CS is .177 both prefix in the same serial no batch CS being in the minority. First S serial no 1 made 6.10.1919 serial 6000 started in 1920. So it is reasonable to assume your was made late october early nov 1919.
At Bisley gun fair today there was a Lincoln Jeffries First batch serial 500 something for £300. unfortunatelly it was just to rough for my liking, too pitted, sanded down finish, and a broken stock but repaired well. Such a shame.
________________________________________
HW97k, HW98, Diana mod 36, BSA improved Mod D, FWB 80, HW45