The inclusion of aligned-spin effects in gravitational-wave search pipelines for neutron-star--black-hole binary coalescence has been shown to increase the astrophysical reach with respect to search methods where spins are neglected completely, under astrophysically reasonable assumptions about black-hole spins. However, theoretical considerations and population synthesis models suggest that many of these binaries may have a significant misalignment between the black-hole spin and the orbital angular momentum, which could lead to precession of the orbital plane during the inspiral and a consequent loss in detection efficiency if precession is ignored. This work explores the effect of spin misalignment on a search pipeline that completely neglects spin effects and on a recently-developed pipeline that only includes aligned-spin effects. Using synthetic but realistic data, which could reasonably represent the first scientific runs of advanced-LIGO detectors, the relative sensitivities of both pipelines are shown for different assumptions about black-hole spin magnitude and alignment with the orbital angular momentum. Despite the inclusion of aligned-spin effects, the loss in signal-to-noise ratio due to precession can be as large as $40\%$, but this has a limited impact on the overall detection rate: even if precession is a predominant feature of neutron-star--black-hole binaries, an aligned-spin search pipeline can still detect at least half of the signals compared to an idealized generic precessing search pipeline. ; Comment: 11 pages, 7 figures; version accepted by PRD