Remove the glass from the tank, screw back in the top piece, now your coil head and coil should be connected to the base again. As stated above, either grab some kitchen glove, wear them, grab the base off the tank and give it a good tuck. If you have no gloves or vaping tweezers, turn off the mod.
Grab a stainless steel nail file, or small thin screwdriver, the little tools you get with every IKEA purchase etc. Anything that's robust and thin/small will do. Now look at the airflow of your tank at the bottom, and slide whatever object you picked through the coil (the little metal slit you can see at the bottom) and get some leverage at the edge of the airflow itself. That will open it without any issues, be advise that the paint of your tank might chip, but at least it won't be stuck to the mod any longer lol