Subwavelength hole array (HA) metasurfaces support the so-called extraordinary optical transmission (EOT) resonance that has already been exploited for sensing. In this work, we demonstrate the superior performance of a different resonant regime of HA metasurfaces called anomalous EOT, by doing a thorough numerical and experimental study of its ability in thin-film label-free sensing applications in the terahertz (THz) band. A comprehensive analysis using both the regular and anomalous EOT resonances is done by depositing thin layers of dielectric analyte slabs of different thicknesses on the structures in different scenarios. We carry out a detailed comparison and demonstrate that the best sensing performance is achieved when the structure operates in the anomalous EOT resonance and the analyte is deposited on the non-patterned side of the metasurface, improving by a factor between 2 and 3 the results of the EOT resonance in any of the considered scenarios. This can be explained by the comparatively narrower linewidth of the anomalous EOT resonance. The results presented expand the reach of subwavelength HAs for sensing applications by considering the anomalous EOT regime that is usually overlooked in the literature.