Check your terminal connections. Wires from the coil enter holes in the forked terminals. After they are soldered, the leads are trimmed. Then the customer solders their wires onto the terminals. If this is done many times, there is a possibility of having one or more coil wires slipping out of their respective terminals, thus causing an open coil.
Use a solder sucker on the open coil terminals to see if this is the case.
Rewinding that coil would be extremely difficult and time consuming and results would be dubious. There are two windings done side by side, consisting of many thousands of turns of very fine wire which is insulated layer by layer with sheets of .0005" Glasine paper. Coils back then we're wound simultaneously, perhaps 24 at a time on a single mandrel. This allows insulation to be put down in very large sheets. The coils are then sawn apart on a band saw. This makes for a very uniform insulation layering. When you try to wind one coil at at time insulation management becomes precarious, resulting in a sloppy coil that we used to call sponges.
Also wire was different back then. Copper content, insulation differences and precise tensioning make reproduction difficult. And the wax impregnate is non existent due to restrictions of the chemicals used. Best bet would be to buy another online somewhere, unless you discovered a wire out of it's home. In that case, you drill out the four corners of the base plate, remove the plate and wrestle with the black tar to get the wire re-soldered.
There was nobody better than UTC at winding those side by side layers of 8000 turns of .044 awg dual section fine wire coils.