We demonstrate the viability of using high-order modes in localized optical resonances combined with gain media to dramatically reduce the linewidths of absorption spectra. Our theoretical study provides a rational design route for small-footprint absorption with a high quality factor, which is typically a trade-off for plasmonic light absorbers. Specifically, we design and numerically investigate a metal-insulator-metal absorber and a graphene Salisbury screen. Additionally, we study the potential application of these structures in sensing in order to achieve higher performance compared with conventional sensors based on the fundamental mode. Our approach, which can readily operate in a multiplexed fashion, has the potential for sensing minute amounts of analytes with a high figure of merit.