Optical microresonators, in particular whispering-gallery microresonators, have proven to be especially useful as chemical sensors. In most applications, the sensing modality has been dispersive; an example is the frequency shift of resonator modes in response to a change in the ambient index of refraction. However, it has been shown that the response to dissipative interaction can be even more sensitive than the dispersive response. Dissipative sensing is most often carried out via a change in the mode linewidth owing to absorption in the analyte, but it has been demonstrated that the change in the throughput dip depth of a mode can provide better sensitivity than linewidth change. Dispersive sensing can be enhanced when the input to the microresonator consists of multiple fiber or waveguide modes. Here we show that multimode input can enhance dissipative sensing by an even greater factor. Having multimode input does not affect the linewidth response, but the enhancement factor for the dip-depth response can be quite large. We demonstrate that the multimode-input response relative to single-mode-input response using the same fiber or waveguide can be enhanced by more than three orders of magnitude. Furthermore, this enhancement is independent of the mode linewidth, or quality factor Q of the mode. The enhancement factor can be predicted by making only two measurements of dip depth in the absence of analyte: one with the two input modes in phase with each other, and one with them out of phase.