magnolia,
Boy, a picture is worth a whole bunch of words. I thought a '58 had a key starter!! I see one wire to the key side of the coil...and I think I can see where it attaches to a small side post on the switch. This would give the coil 12 volts in "stomp" mode.
I wonder where the coil gets its power in "run".
Is that oil from the valve cover leaking onto the coil, or has the coil started leaking? Don't forget that the coil might have a part in the hard hot start condition.
If you don't have a ballast resistor mounted somewhere, you need a coil that indicates "no external resistor needed"..hard to see on an old dirty coil!
Stuart