I had a similar problem to the OP concerning a garage with too much stuff and the rear view mirror. In my case I was walking between the cars and bumped the driver’s side mirror on the 1993 Oldsmobile we owned. I didn’t really feel any pain from the “collision”, but when we drove the car, the mirror jiggled so badly that the image was terribly distorted rendering the mirror virtually useless. Something was damaged in the mechanism that controlled the mirror and the whole mirror had to be replaced. Another time I was descending the fold up stairway from the attic of the garage. I missed a step and fell against the car, leaving a,small dent in the front fender. I was unhurt. Now we back both cars out of the garage to protect them if I am going to do anything in the garage.
In the OP’s situation, maybe they still carry hand adjustable mirrors at auto parts stores or J.C. Whitney for around $10. These mirrors are attached to the body with metal screws. Outside of having to roll down the window to adjust the mirror and not looking like original equipment, I think it would be a better solution than taping on the mirror.
Back in the " good old days" cars didn’t come with outside rear view mirrors. I had to put an outside mirror on the 1965 Rambler I bought in 1965 and for all the earlier cars I owned. The outside mirror was not standard equipment.