Glad I could help, at least with getting the pictures posted. I have a couple of White machines, but haven't had them out for a while. I don't remember them having 3-cord zip cord. I'm thinking that an electric foot controller would work. I know they do for old carbon pile controllers and pretty sure rheostat controllers.
Is the cord long enough to go to a foot controller? I'm wondering what the wiring looks like after it goes under the machine. I know I have seen a bunch of configurations for the wiring. I have also heard of people thinking that it didn't have a foot pedal, but finding that it was wired to a knee controller. Some machines use a knee controller that is actually a foot controller in a bracket in the cabinet. Unfortunately there isn't a picture of the motor controller, but two motors for a Dressmaster at
Louise .... or the new Singer ....... no choice at all. one is blade type and the other is a pin type.
Now that you know how to post pictures, maybe show the ends of the wires that you have.
ETA- the place that I and others used to refer to for vintage machine parts is no longer in business. I have picked up controllers from thrift stores and have switched out if had the cord with the correct plug.
Janey - Neat people never make the exciting discoveries I do.