I installed header pins that fit into a feather microcontroller, but since they are not double sided header pins, the top side is too short to attach normal jumper wires to. Is there some clean, non-permanent solution to this where I can still attach jumper wires, but not solder or desolder any existing pins?
Either cut the plastic and lift it off the pins, or use test clips or micro grabber clips to attach to the remaining headers. You could also use wire wrap. Or solder a second header to that header. It's possible just ugly.
Or save the hassle and just desolder and resolder a new one of the right length. It will take you less time than hacking something.
Desoldering seems to be the easiest.
These pins are very easy to desolder: grab them one by one with needle nose pliers, heat the other side, and pull. Once the pin is heated, it will soften the plastic and slide through with very little force. You don't need to hack the plastic. This will take like 2 minutes.
These pins can be connected to jumper cables just fine if jumper cables have the right kind of connectors. Remove the black cover to see what connector you have.
The good variety can be connected to the pins you have. The one in the middle will be on its limits, but may hold in place if you don't pull on the wires. The bad variety won't hold on anything but a full-length pin, plus it only works well once.
With the bad kind, it's sometimes possible to push the springs further down to make it temporarily better, but don't hold your breath. If you can't get your connectors to hold in place, your next solderless option is the clamp wires:
These take much more space than regular connectors and are quite unstable, but can hook up to very small pins.