A few sources for very fine wire are:
Old relays. The wire can be in many different gauges depending on the voltage & current needed to activate the coil.
For heavier wire, the starting solenoid for a car starter, an old electric motor, or a generator windings can be used, as well as disassembling old power transformers. Bad transformers with open primaries can usually provide a lot of good wire too.
Vertical output, Horiz. output, interstage and audio transformers, yokes, degaussing, and field coils from old CRT TV sets are another good source.
(I must note here that many CRT & computer monitor sets went to Aluminum for yoke coils in the 90's. You can usually tell by the weight of the yoke, with some practice.. Also in the late 70's/ early 80's many mfgrs. went to aluminum wire for the Degaussing coils.)
Microwave Oven power transformers are usually a good source too. I have only seen a very few of them that used aluminum wire. Note: Many of the newer oven are using an SMPS, not a power transformer. EZ way to tell is if you lift the oven & it feels really light, it likely has the SMPS supply in it.
The transformers and coils from the computer SMPS often have a good amount of copper wire in them.
Hint: Need a good size loop antenna for a DF rig, or just to increase your radio's signal? Use the degaussing coil from an old CRT type TV set or computer monitor. Just hang it on the wall. It will be very directional. The radio won't are if its aluminum or copper, either.
For a long wire antenna, just carefully remove the tape on the coil and unwind it.
I have lost track of how many long wire antennas I made from degaussing coils.
The back light transformers from the flat screen TV sets & monitors can provide some "smaller than a hair" thin wire too.
Old CRT type monitors and TV sets can generally be found for free at garage sales.