I'll talk about the mirror housing and the mirror itself just for reference for others.
The mirror housing is tightened with the screw that's on the back of the arm that attached to the door. Either the screw is loose or you're F'd because it's held tight against the arm with a ball mount. If you over tighten the screw you can crack the casing of the mirror housing that the ball seats in. See the red arrow.
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The mirror itself actually rotates around another little ball that plugs the hole the piece that mounts the housing to the arm went through.
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The 2 little sticks sticking up on either side of the hole are bump stops for the mirror. The mirror itself is held in position by the spring in the remote lever. My spring and remote housing was was just old and worn out, and where the arrow is was cracked on the back side which didn't help the spring either.
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When I went to disassemble, the cheap pot metal just crumbled. which is one of the reasons I purchased a repo mustang remote mirror, so I could use the controller (spring and housing) as a donor new piece. I removed the controller and tossed the rest of the mirror.
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It was complicated to remove the old cables and correctly attach them to the new controller (although each of the 3 cables was marked at the factory with a different color on the tip). I had to cut the little brass bushing off the ends, thread the cables through the housing and the controller, then use a special tool to crimp new tiny brass bushings on the ends. The repo controller worked out great. The spring is tight which keeps the mirror in place, and it perfectly fit the old bezel nut and handle that I had re-chromed with the mirror.
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That's the long answer. For you it's most likely a worn out spring in the controller. I'm not sure how you can tighten it up without replacing it like I did. You may be able to somehow shim your spring at the bottom. If you're local to the Phoenix area I'm more than happy to help you graft a repo controller on since I had to buy the tool and a bag full of bushings. The mirrors are rare so don't toss it if you swap it out.
Sometimes North East Ford Parts has a used remote mirror for sale (they have one listed now for $149) but the pitting is really bad. They always seem to be which is why I just rebuilt and re-chromed mine.