Starman was in a 6 hour holding orbit before the injection burn to Mars (Assuming that happens at the 6 hour mark). This was done to test the capability to put a satellite directly in GEO orbit. Did Starman actually orbit the Earth multiple times in that period of time?
As I could understand, there were two additional burns after the orbit was achieved. One, in the first orbit, that raised the Apogee to 7000 Km. The other was, probably, done in the Perigee again, and was the one that would put Starman in the Heliocentric orbit that stretches from Earth's to Mars' orbits. I think they opted to separate into two burns so they could take advantage of the Oberth Effect.
I would, then, say that it has completed only (roughly) two orbits around the Earth. The 6-hour wait was necessary due to the higher orbit period after the first burn.