new iteration, this time with the old two magnet approach, but with a flux ring surrounding them:

Both the oldest and this version of the sensor give me a greater sweep over the raw values read from the joystick, but this might be just the result of using two magnets. I tried to make the ring out of the steel spring from a windscreen wiper, but I realized that the specific steel didn't have much permeability. And annealing it brought its magnetic permeability down to near zero! Looking things up on Wikipedia I found that the magnetic properties of annealed steel vary a lot between different steel grades. The ideal material would be mu-metal, a steel that contains a certain percentage of Bronze and whose permeability increases when annealed