DRV5056: Hall sensor recommend request

Thank you for reaching out with your question here.  It appears that there are 4 natural states for this joystick.  Rest, Depressed vertically, Depressed and pushed up, or Depressed and pushed down.  

To detect this motion with a single sensor will require the use of a linear sensor, and the best option will really depend on the placement, magnet material, and magnet polarization.  From the materials, I didn't see any reference to the direction of the magnet polarization, but this field calculator tool may prove useful for this

https://magnetrechner.de/en/field-as-required-in-space/ring/axially-magnetized

If needed a ring magnet with axial, radial, or diametric polarization can be selected.  

A 1D sensor (DRV5055,DRV5056,DRV5057) in the position indicated would be sensitive to only the component of the vector which is directed through the package in the z direction.  For these devices a field directed from the bottom to the top would be considered positive.  

In the drawing it appears that the sensor is located roughly halfway between the inner and outer diameters of the ring.   Here, for an axial (in z direction) polarization, I would expect all of the field to be directed in this direction, though the sign will be determined by the direction the magnet is facing.  We would expect to see a shift in magnitude when the joystick is depressed.

The challenge I would expect come when the joystick is then shifted upward or downward.  In either direction, I would anticipate that the vector component would remain mostly in the same vertical direction, but there would now be a lateral component as the field begins to wrap around the magnet.  I expect that there might be some slight difference between the magnitude observed for equal shifts in either direction, but this may prove challenging to distinguish.  I would recommend carefully considering the placement using the field calculator.

An alternative would be to use a device like TMAG5170-Q1, which is sensitive to the entire field.  With sensors aligned to each X,Y, and Z, we should be able to reasonably detect all 4 positions.  I would anticipate seeing a magnitude shift for Z, and then either X or Y (depending on placement and rotation) would shift positive and negative for the remaining positions.