I realized that other neighboring countries may still have used the route thou the desert, even if it was not called Overland Mail Baghdad-Haifa anymore.

One of these countries apparently has been Jordan, or better said Lebanon as the cover shown here proves. The cover was mail from Jordan on 5 September 1951 and was sent via Beyrouth (7 September 1951) to Baghdad (11 September 1951). Transit and arrival postmark on reverse.

It would be interesting to know who the Transport Company was which was entrusted in carrying mail from Lebanon / Syria to Iraq at that time.