I could be mistaken, but I seem to remember seeing a photo or video frame of an officer examining a jacket at the scene where it was found (the parking lot behind the building on Jefferson). But if I understand your theory correctly, LHO would have left CE 162 in Irving.
It's not my theory, it's merely a possibility. What we seem to have is two witnesses who contradict eachother. In my opinion, it's a matter that the WC should have resolved, but never did. They just ignored Buell Frazier's testimony and took Earlene Roberts word for it. Not really the way to conduct an investigation, if you ask me, but that's what happened.
There is indeed a photo of an officer holding a jacket, flapping in the wind, at a parking lot near the Tippit scene. That jacket was found, under a car, by an officer who was never identified and it was called in to the DPD dispatcher as being white. There is more to that story as well, just as there is to all the other pieces of physical evidence involved in the Tippit murder.