Thanks for the lock Gerd. Sorry for the slow reply, but I have been away from my computer all afternoon.
The source I was using when I made my initial post cited Linz/Schrader regarding the only surviving Rex-Simpex after the Elite takeover as the 13/40 hp which was sold as the Elitewagen 13/40 PS. This source made no reference to any other post-war Rex-Simplex models. Despite the lack of Rex-Simplex information, I have managed to find a second internet source (I have no idea regarding its reliability) which says:
"After the war were updated two designs from before the war, the 10 / 28hp and 13 / 40hp.
In 1921 the company was absorbed by the "Elite Motorenwerke AG". Over the next two years until 1923, it continued to produce a single model, the 13 / 40hp (3176cc)."
So, if the puzzle car is a post-war Rex-Simplex and the above information is correct, but it is not a 13/40 hp, it must be a 10/28 hp.
As for the model designation I can find nothing, so I can only speculate. The puzzle car appears to have six seats, although the middle two may be jump seats. In earlier times I would suppose this type would have been called a Phaeton (either Double Phaeton if it has jump seats, or Triple Phaeton if it has permanent seats). In the early teens the term Phaeton tended to fade in favor of the term Torpedo, which in the post-war period (at least in the US) tended to morph into the term Touring Car, or Pullman (or Limousine) Touring Car if it has extended length for extra seats. Based on this I would suppose that this vehicle would be marketed as a: 1921 Rex-Simplex 10/28 hp 6 seat Touring Car, or perhaps: 1921 Rex-Simplex 10/28 hp Pullman Touring Car.
