Electrical Demon?
What kind of ignition do you have, IE Mopar ECU, HEI, points, what, exactly?
The power you get to the coil in "run" is NOT what "starts" the car. There is a separate contact in the ignition switch which feeds what is known as the "IGN2" circuit or the "bypass." This is one brown wire comes out of the switch, goes through the bulkhead, and goes to the coil + side of the coil.
BUT WE NEED to know what you have for an ignition, and what YEAR is the car?