I found
this youtube that shows a very simple 3-transistor circuit for tracing AC lines in walls. The video starts out about as basic as you can get, just skip past the dumb stuff to the schematic and showing its operation. Basically a capacitively-coupled 3-transistor Darlington running an LED. So this one is passive, in that it's just listening for 60Hz. That would be OK for the circuit that drives the relay coil (or up to the transformer, if there IS one). Tracing the circuit on the other side of the relay with this would require running some live AC up to that screw-in adapter, but runs the risk of connecting the hot side to the local ground.
Another approach I found used a simple AM-band oscillator modulated at around 350Hz with a 555 astable oscillator,
here. That way you can use an off-the-shelf AM receiver. It uses some parts your average machinist won't have, but, knowing you, you may....or something equivalent. A simple ring oscillator built with CMOS or TTL inverters would do the job, and probably would be the first thing I'd try. Probably not too stable w/respect to temperature or supply voltage but for down & dirty...
Finally, off-the-shelf circuit tracers seem to be pretty cheap, enough so that they likely can be found in one of those big-box stores. It all depends on how much time you want to spend vs. money you want to spend.