If your coil was oscillating, then perhaps an iron core moving through it would cause perturbations which are detectable. But that would require extra logic to compare the expected oscillation frequency with what the coil is actually oscillating at.
Since you say that tilt switches are not an option – for reasons I’m not entirely sure I understand – another option is to have a linear Hall effect sensor mounted nearby a small magnet. If the magnet moves relative to the sensor, then that is a change which can be acted upon. A linear sensor makes it possible to use a trim pot to tune the sensitivity.