But did he?
After three or four shots at close range?
Would he need to reload?
With the greatest respect, 'to reload' is a crap answer.
Try checking the facts ... "To reload" isn't a crap answer, because it's what Oswald demonstrably did.
He left his rooming house with the .38 revolver in his jacket pocket. He was intercepted approximately a mile from the rooming house by Tippit - a patrolman identifying Oswald as a figure consistent with the BOLOs that had been broadcast multiple times since the Kennedy shooting in Dealey Plaza.
The evidence indicates Oswald fired 5 times during his encounter with Tippit. Four bullets hit Tippit, and the mismatch between the number of cartridge cases recovered at the scene and the bullets retrieved from Tippit's body indicates a fifth one had been fired but missed Tippit.
Regardless of what had happened at Dealey Plaza, Oswald was acting furtive and agitated. He'd just been stopped by a cop, whom he unexpectedly shot. As he walked away from the shooting scene, multiple witnesses testified he was shaking the revolver as if to remove the spent cartridges. That particular S&W revolver had a sliding ejector. Oswald apparently didn't initially use it, because the spent cartridge cases would remain in their chambers without either using the ejector or manually removing them (e.g., by shaking the pistol).
He'd shot a cop with multiple witnesses on the scene, so the shit had most definitely hit the fan. He'd already demonstrated his inclination to meet police inquiry with gunplay. Reloading his only weapon would have been a top priority.
When apprehended at the Texas Theatre Oswald still had the revolver. He pulled it out from his waistband and attempted to shoot one of the arresting officers, who grabbed the pistol in such a way as to prevent the hammer from impacting the chambered round.
The revolver held 6 live rounds, and there were an additional 5 loose cartridges in Oswald's pocket.
He'd reloaded.