What 22 AWG wire to use for wrapping?
Making 22 ga wire wrap connections is not something I'd want to try (and I did wire-wrap for decades - still do, sometimes). As has been mentioned, you'd need a special bit for a wire wrap gun. But that's not the reason I'd avoid it. Wrapping solid 22 ga around a .025 square post will require a lot of torque to make the bends small enough. I'd worry about keeping the tool straight as the wrap process occurs.
Instead, I'd suggest you simply solder a piece of 22 ga hookup wire lengthwise on the pin, then slip a piece of heatshrink tubing over the joint and apply heat.