Had this problem on a customer car in the late fifties, confirmed it on test drive. Car would start right back up. After looking at all the usual things we had him leave the car. He had to take his the ignition key off his ring when I dropped him off, as he had all his work keys on the ring. When driving back to the shop the problem was gone. Found that the switch was worn probably due to all the weight of the keys, solution was a second ring with only the keys for the car. Some times the basic diagnostics don't work.