I designed this simple box to go with the Heat Insert Press by BaGooN. It fits in between the legs of the stand and has ridges left and right so you can lift the stand and box together at the same time. The box has three compartments for heat inserts with a curved bottom for easy picking. There is also a row of slots to hold the press bits for the soldering iron. I use it to keep everything together and ready for use. The design is simple  (Tinkercad) and not pretty, but it's functional.

Click here for the Heat Insert Press by Bagoon: https://www.printables.com/model/808502-heat-insert-press