Oswald was last seen at the Ballew's Texaco Service Station and was next seen 15-20 minutes later half a mile away by Johnny Brewer, so obviously Oswald stopped somewhere, gathered his thoughts and if needed he cleaned himself up, Oswald did what was necessary to not look like a suspect just like when he discarded his jacket.
...or it wasn't the same guy.
FEDERAL BUREAU OF INVESTIGATION
Date 1/22/64
Mrs. MARY BROCK, 4310 Utah, Dallas, Texas, advised that on the afternoon of November 22, 1963, she was at the Ballew Texaco Service Station located in the 600 block of Jefferson Street, Dallas, Texas. She advised that at approximately 1:30 PM a white male described as approximately 30 years of age; 5 feet, 10 inches; light?colored complexion, wearing light clothing, came past her walking at a fast pace, wearing a light?colored jacket and with his hands in his pockets.
Approximately five minutes later two individuals from Johnny Reynolds Used Car Lot, 500 Jefferson Street, appeared at Ballew's Texaco Service Station, making inquiry as to whether she had noticed the young white man come by the station. She indicated she had, at which time they informed her that this individual had in all probability shot a Dallas police officer. She advised she informed them that the individual proceeded north behind the Texaco station and she last observed him in the parking lot directly behind Ballew's Texaco Service Station.
Mrs. BROCK was shown a photograph of LEE HARVEY OSWALD, New Orleans PD 9 112723, dated August 9, 1963, which she identified
as being the same person she observed on November 22, 1963, at Ballew's Texaco Service Station.
Mrs. BROCK advised at the time she saw OSWALD on November 22, 1963, she was unaware of the fact that President JOHN FITZGERALD KENNEDY had been assassinated, and she was unaware that Dallas Police Officer J. D. TIPPIT had been shot.
on 1/21/64 at Dallas, Texas File # DL 100-10461
By Special Agents JOHN T. KESLER and VERNON MITCHEM - LAC Date dictated 1/22/64