Current searches for the gravitational-wave signature of compact binary mergers rely on matched-filtering data from interferometric observatories with sets of modeled gravitational waveforms. These searches currently use model waveforms that do not include the higher-order mode content of the gravitational-wave signal. Higher-order modes are important for many compact binary mergers and their omission reduces the sensitivity to such sources. In this work we explore the sensitivity loss incurred from omitting higher-order modes. We present a new method for searching for compact binary mergers using waveforms that include higher-order mode effects, and evaluate the sensitivity increase that using our new method would allow. We find that, when evaluating sensitivity at a constant rate-of-false alarm, and when including the fact that signal-consistency tests can reject some signals that include higher-order mode content, we observe a sensitivity increase of up to a factor of 2 in volume for high mass ratio, high total-mass systems. For systems with equal mass, or with total mass ∼50 M⊙, we see more modest sensitivity increases, <10%, which indicates that the existing search is already performing well. Our new search method is also directly applicable in searches for generic compact binaries.