The solution is to replace the grid wires with good
quality shielded cable. Completely remove the old wire
by fishing it through the holes in the tag board. Route
the shielded cable above the tag board. Connect the
shield to ground on the jack/pot end; at the tube end,
cut off the protruding shield conductor and shrink wrap
it; don't connect it to anything.
 
I highly recommend this mod for blackface Fenders.