[morzh]

The best way to me is: (I rewind coils this way):

1. Make a sketch. Measure the length of the winding along the bobbin, position relative to other windings. Note the way the wire goes to the first pin (choose one to be the first pin, does not matter), with the orientation of the coil noted (say, on the sketch pins to the left, top to the right), then start unwinding the coil, counting turns. When finishing the last turn, note the direction of winding, put it on the sketch.
Measure or estimate the wire gauge. If litz is used for the wire, use litz, not a single wire, it matters.

2. Connect your wire to the pin that you want to start winding from. Do not solder, just wrap the wire around it. Wind the section, as close to the original as posible: wire gauge if many turns and the coil is used in a resonant tank (if two-three turns, like in RF non-resonant transformers, it is not important as much as long as it fits, in resonant coils, lie RF winding or antenna coils that connect to tuning caps it is important), length of winding (same criteria as above), deinitely direction of winding and connection to pins.
Keeping the winding taut, thread the wire and wrap around the other pin.
Secure the winding with whatever used for that (I use fabric glue).

It takes 20 minutes tops.

3. Unwind the ends from the pins, clean and solder.