And how do you explain FBI Bob Barrett saying that it was a wallet?
My opinion is that Barrett (who went to the Texas Theater and then straight to headquarters after Oswald's arrest) saw Oswald's arrest wallet while inside Westbrook's office back at headquarters. This is where the arrest wallet was for a while. Barrett said he went there to assist with the arrest report. I believe this is when Westbrook approached Barrett with the Oswald/Hidell names; back at headquarters.
I believe this for two reasons.
First, there are no reports filed by Westbrook or Barrett of any wallet being discussed at the scene at Tenth and Patton. Barrett filed a report that weekend, but never mentioned any wallet. My opinion is that if a wallet had been discussed at the scene with Oswald/Hidell identifications inside, it would have been mentioned in such a report.
Second, my opinion is that if a wallet was found near Tippit's body (or the patrol car), then the name(s) of the wallet's owner would have put out over the police radio to the dispatcher for the purposes of placing an APB on this person. Just like no wallet mentioned in any of the reports of those "involved", no APB was ever put out over the police air waves.