The cord that you're referring to is what I call "tie wire", because it ties the individual springs together. Its proper name is "stake wire". It is paper covered oil impregnated wire. Any upholstery shop should have it in stock. It can also be used for borderwire ( around the perimeter of a seat spring assembly) instead of 8 gauge steel wire. It also is used embedded in foam seat buns to hog ring listings and wires to define an insert section of a seat cover. The first Mustang front bucket seats ( '64 1/2 through '68) were done this way, and probably your Galaxie seats.
No one lives forever, the trick is creating something that will.