Best way to terminate fine gauge polyurethane magnet wire
To strip the fine wires (AWG32 is not all that fine) with polyurethane insulation you can usually use a small solder pot. Set it a bit hotter than you would a bench soldering iron. If you use one of the very inexpensive ones, it helps to put a small thermocouple into the pot to monitor the temperature. A bit of peanut oil on the top will  prevent too much oxidation from occurring. Something with a 1-2" (25-50mm) inside diameter is adequate. 

If the insulation won't burn away, you can use mechanical or chemical stripping. 

Typically you'd tape some sturdy insulation material to the outside of the finished coil, splice the wires, insulate the joint if necessary, and then wrap more tape around the outside to secure the joint. Leave adequate slack on the fine wire so that it won't be drawn taut by thermal expansion.