And should I solder it to remaining stub on the antenna?
Use a flex-able wire in-between, just the weight of a coat hanger would sooner or later tear the pad and or solder off the PCB.
If using the coat hanger, and you don;t plan to move it all around, I'd try and secure it nice and solid to the stereo 1st. Maybe run it down as deep inside the case as it can go, hopefully to the bottom, or tape on on the outside, nice and solid, then run your little wire out to it and solder on where ever. But what ever the order, I'd want it nice and solid.
If you plan to move the ANT. around, just use extra wire, and try to have the wire near the solder joints, not move at all.