Fixing broken Shure 545 (Unidyne III, same as SM57)... argh!

Argghh...

One of the Shure 545 (Unidyne III, predecessor and very similar build to SM57) mics I recently acquired has a broken coil-

In the part under the grille assembly where the wire comes out from under the diaphragm and goes down the side of the mic (under some tape), the coil-wire is broken. I believe I should be able to pull a tiny bit more through from the coil and resolder it carefully, but I'm not sure if there's any pitfalls to look out for.

Anyone got any suggestions, or experience fixing broken coils on 57s?

Thanks
Jon