I’m a new member from the UK & have recently bought the above. Can anyone tell me how to repair/rebuild so that they stay solid when adjusted? When I do the screws up at the back of the housing the actual mirrors are still floppy,
I would think one of two things is occurring. If the screws that you are using to tighten the mirrors are too long they will bottom out and feel tight before they make contact with the mirror bracket and then cannot hold the mirror itself tight. The solution here would be to purchase new shorter screws (am not sure of thread count difference and screw size differences between the US and the UK) or to very carefully hacksaw the ends of your current screws being sure not to foul the threads. This can be tricky as the screws typically are fine thread for this application. Take real care if you attempt this. In many instances a washer would do the trick but I think the screws are countersunk and a washer cannot be used.
The other issue could be threads that are stripped either in the mirror itself or in the screws. That would require new screws and a tap and die to make new threads in the mirror itself.
One other solution would be to purchase new mirrors. Many of IMOA’s suppliers carry mirrors that would be appropriate for your car.
Good luck with your car. ’62s are pretty rare (love the taillights) but are gaining in popularity especially the convertibles.