Good grief!
"Oswald stated he then went home by bus and changed his clothes and went to a movie" (Bookhout/Hosty report on first interrogation 11/22/63)
Whatever Mr Oswald really did tell them, they kept evolving the story they put in his mouth so as to fit the story they were writing and modifying. If that lent the impression of a suspect who kept changing his story, all the better. Mr Oswald in fact made a simple claim and stuck to it across all of his interrogations. That simple claim being known to be veracious AND disastrous for the Tippit timeline, it was suppressed.Le did NOT "keep changing his story".... In fact he simply said that he went home by bus...That wasn't totally accurate but I don't believe that it was intended to be a blatant lie.... Lee simply didn't see the point in describing the details of how he traveled to Oakcliff.
And to be totally honest ...he MAY not have wanted to reveal that he was in a hurry to get to the theater, because he wasn't going to the theater to simply watch an old war movie. He was going there to meet his handler and start his sojourn to Cuba.
If he had told Fritz that he had hired a Taxi to take him to his room, naturally Fritz would have asked him why he was in such a hurry ......and why did he leave the area at the TSBD when there was so much mesmerizing activity happening there.
But back to the basic ..... Lee DID NOT
"keep changing his story" When Fritz asked him if he had ridden in a taxi, he readily admitted that he had in fact taken a taxi to the rooming house.