Wire wrapping - how to attach a breakout board without soldering headers?
As I understand your question you are not opposed to using some kind of header pins, you just don't want to solder them to the breakout board. It's difficult to get a good connection between the pin and the board without soldering, but a press fit pin might work for you. The pin shown below is from Vero Technologies. You can wrap on both the top and bottom of the pin. The pin must very tightly in a plated-through hole, and I used to have a spring-loaded tool that would insert them.