We propose a scheme to extend the measuring range of a transverse displacement sensor by exploiting the interaction of an azimuthally polarized beam (APB) with a single metal-dielectric core-shell nanoparticle. The focused APB illumination induces a longitudinal magnetic dipole (MD) in the core-shell nanoparticle, which interferes with the induced transverse electric dipole (ED) to bring forth a transverse unidirectional scattering at a specific position within the focal plane. Emphatically, the rapidly varying electromagnetic field within the focal plane of an APB leads to a remarkable sensitivity of the far-field scattering directivity to nanoscale displacements as the nanoparticle moves away from the optical axis. Moreover, the scattering directivity of the APB illuminated core-shell nanoparticle is also a function of structure-dependent Mie scattering coefficients, rendering the measuring range of the transverse displacement sensor widely tunable. The culmination of all these features enables the continuous tuning of the displacement measuring range from several nanometers to a few micrometers. Thus, we envision the proposed scheme is of high value for modern optical nanometrology.