From your description the thin wire is probably the low tension cable from the coil to the distributor. Depending on the age of your Morgan it may have a 'contact breaker' in the distributor, that is connected to the coil via the piece of wire that you describe. The fault may be a bad connection somewhere in the wire (most likely at the connector at one or the other end of the wire) or a faulty connection within the coil (from the 'bolt' to the copper wire of the primary circuit within the coil). If you apply pressure to the 'bolt' (stud) on the old coil do you notice any movement, is it loose ?
The fault should be easy to trace using a multimeter. The resistance between the two bolts (studs) on the old coil should be just a few ohms. If you apply pressure the the 'bolts' on the coil, while measuring the resistance it should stay constant.
The resistance from end to end of the wire from the distributor to the coil should be effectively zero ohms. Again you can apply stress to the cable while measuring the resistance to see if there is any change.
In the absence of a clinical test with a multimeter I guess the practical test is to start the car and wiggle that thin wire to see if the problem still exists
