Our 1929 Chicago-style bungalow has a two-section staircase joining the main floor and the upstairs - six stairs up to a landing, then turn 180 degrees, then seven more stairs up.

I'm trying to re-mount our handrails as they were loose when we bought the house. The problem is that the wall between the stairs is only about 36" wide, and I can only find one stud directly in the center. I'm using a very powerful rare earth magnet which has been effective at finding studs.

How do I safely mount a handrail when there is only one stud in the wall, and that in the worst possible location?