Extending header pins
Figure 1. Double-sided header pins.

Get some of the double-sided header pins and solder these to the underside of the board. It may be a bit tall but electrically will be OK.

Cut the strip to length, tin each of the pins, line it up with the pins on your PCB and tack-solder to the highest pin on your PCB. Then connect the end pins to the PCB pins. Finish off the rest.

It may be stronger and easier if you mount the extensions behind the existing pin stubs so that they're almost touching the board. That way you will be joining them along along parallel faces rather than just at the ends.

Regarding your soldering skills, you can aspire to this:



Source: Little wire dead-bug art.