For winding custom coils, or very fine-pitch rework, we always reach for magnet wire. Magnet wire is very very thin (0.1mm diameter!) copper wire with a very thin layer of insulation. It's used for a range of electronics, hacking and making applications. Commonly these are used for winding transformers, inductors, motor or solenoid coils, speakers, hard disk head actuators, electromagnets, pickups, among other things. However, this wire is also really good for when you have to do PCB repair, jumper wires, or solder directly to SMT pads.
This spool is ~11.5 meters of ~28 AWG thick magnet wire, just enough to do a few projects. We recommend a spool for every engineer's workbench. Like all magnet wire there is a thin enameled insulation. Some people like to remove it with a small flame (from a lighter, say) but we've found that an easy way to remove it is to melt a blob of solder onto your iron and then dip the end into the blob for a few seconds. Remove the wire and it'll be both cleaned off and tinned!