Hi Axel,
Don't connect a ground wire yet, first we want to figure out what the source of the hum is.
Check that the nut holding the 1/4 jack is secured well. Don't go crazy tight, just make sure it isn't wiggling around.
Next, open the backplate and see that the pickups are plugged in well. You might just want to unplug each one and the plug it back in sercurely.
It would be useful to know which pins of the 5-pin make the hum happen. You can describe it as a smile from left to right or send a drawing.