In the book, Hunting Weapons, by Howard L. Blackmore at plate 138 there is a picture of an almost identically stocked Rigby .303 Mauser made for a one-armed British Officer.
On that example there is a small leather covered cheek rest at the turn of the stock wrist and the “paddle” is leather covered presumably to be held in the armpit by the stump of his arm.
It was then in the Tower of London, but will now be at the Royal Armouries at Leeds where Jonathan Ferguson or Mark Murray-Futter should be able you give more details.
Does yours have a Rigby number on the bottom of the trigger guard?