My guess is that he was just trying to make sure that Tippit was not going to recover and identify him later on. Picking up Tippit’s gun would have potentially incriminated him (in case he got caught with it, or left a finger print on it).
Well at least you recognize that the shooting of Tippit was an EXECUTION....... I believe it was premeditated. It does not appear to be the act of a fleeing fugitive.
There can be no doubt that the bullet in the brain was intended to ensure that Tippit was dead..... A fleeing fugitive probably wouldn't have wasted time in clearing out of the area as soon as Tippit was down. The FACT is the killer did NOT hastily dash away from the scene, indicates that he intended to execute Tippit.