Anyone have any clever tips or tricks to snaking a wire on the inside of a wall?

I'm adding an outlet underneath a cabinet so I can hide all equipment and wires. I've cut a hole for the new outlet, cut off the circuit I'm extending and trying to snake the extra length of wire down.

It's only about 3 feet, but off center (not directly below the existing outlet) In hind sight I would have tried directly below to eliminate that aiming. But on top of it all, it's an outside wall so I've got insulation to deal with. And I've just been unable to find the other end/hole.

I just wondered if a stud finder would pick up the wire so I could trace where it actually is? My solution in the past was always just make my hole larger and if I could avoid that I'd like to.