How to I wire this particular variant of 2.5mm female jack?
As you insert a plug into the jack, some of the pieces of metal inside will move along the outside of the plug, and some will remain stationary. The pieces that move are the ones you need to solder the signal to, and the others are connections to the moving pieces when there is no plug in place. The specific application will dictate whether or not the stationary pieces should be soldered to.