I just recently did the same exact thing. I installed a Homelink, Compass, Temp, auto dimming mirror.
The connector that is just hang there is the factory connector for the dimming mirror. The new mirror you have uses a different connector and has its own harness. The reason they can't tuck it up into the headliner (or at least easily) is the wires for the compass pod come down the same harness. The compass still needs to remain plugged in to feed your info center. If you were to unplug it you get a constant error on your info center display.
What I did with mine is took the connector apart and fed the factory wires up thought the loom and brought my homlink wires back down through. My factory wires and connector are tucked up in the headliner. Mine looks stock, but it took a lot of work to do the wires that way.
By the way, I hate to admit it, but I ALSO broke the windshield trying to remove the factory mirror. I even had the special Ford tool, it just would not come off, next thing ya know, CRRRAAACCCKKKKK. The new windshield came from Ford with the Mirror Nub already glued to it. So my near $400 mirror cost another $250 after the insurance deductible. Blech.
If you take a close look at it, and are handy you could fix it, but again... it is a bunch of work... and you need to be handy with wiring and be capable to take the connectors apart. If you don't plan on putting a stock mirror ever back in it, you could clip off the connector and tape up all the ends of the wires, tuck them into the loom, then they won't bother you. Be careful though, you do have +12V and Ground, along with +12V when in reverse... if you let those wires ground out, you will blow a 10amp fuse down near the drivers floor well fuse panel. (I know because I blew it... LOL)
Good Luck, sorry for the book.