I found I had to really wiggle my charger wire around to make good contact, no matter what charger wire I used.    The lighter works fine but there seems to be too much room inside.

We'er discussing a 7.00 replacement part!! Screw out/screw in. Prolly your best bet.
6sal6

I agree with 6sally6.  And I think the part can be bought for about $4 instead of $7, if it matters to anyone.

The good part is the tubular part of the charger and its internal connections are pretty much generic.  You can buy a cheapo replacement at your local auto parts store and use only the barrel part.  Make sure you find one that has the rolled threads on it so the Mustang bezel will attach to it.

Since you know your wire is hot, just plug it into the identical stud on the back of the new one. 

In my 66, it has the same problem.  It has power but no connectivity.  Sometimes a charger will work but most often not.  Since I take the car on long trips, it finally dawned on me that I need more than one charger receptacle since there are likely TWO cell phones and one Garmin to be plugged in somewhere.  I paid the $6.95 and purchased a dual receptacle for under the dash.  Doesn't look very original, but it does the job and can be easily removed if I ever get tired of looking at it.

Follow up:
I took it apart, applied some steel wool and a little elbow grease and everything works now. It's an original with Cuno Meridian, Made In The USA. Fifty years old and still red hot! Cost, 50 cents for the wool and satisfaction of working in the garage!