Optical microresonators have proven to be especially useful for sensing applications. In most cases, the sensing mechanism is dispersive, where the resonance frequency of a mode shifts in response to a change in the ambient index of refraction. It is also possible to conduct dissipative sensing, in which absorption by an analyte causes measurable changes in the mode linewidth and in the throughput dip depth. If the mode is overcoupled, the dip depth response can be more sensitive than the linewidth response, but overcoupling is not always easy to achieve. We have recently shown theoretically that using multimode input to the microresonator can enhance the dip-depth sensitivity by a factor of several thousand relative to that of single-mode input and by a factor of nearly 100 compared to the linewidth sensitivity. Here, we experimentally confirm these enhancements using an absorbing dye dissolved in methanol inside a hollow bottle resonator. We review the theory, describe the setup and procedure, detail the fabrication and characterization of an asymmetrically tapered fiber to produce multimode input, and present sensing enhancement results that agree with all the predictions of the theory.