The recent detection of the binary-neutron-star merger associated with GW170817 by both the Laser Interferometer Gravitational-Wave Observatory (LIGO) and Virgo and the network of electromagnetic-spectrum observing facilities around the world has made the multi-messenger detection of gravitational-wave (GW) events a reality. These joint detections allow us to probe GW sources in greater detail and provide us with the possibility of confidently establishing events that would not have been detected in GW data alone. In this Letter, we explore the prospects of using the electromagnetic (EM) follow-up of low-significance GW event candidates to increase the sample of confident detections with EM counterparts. We find that the GW-alert threshold change that would roughly double the number of detectable astrophysical events would increase the false-alarm rate (FAR) by more than five orders of magnitude from 1 per 100 years to more than 1000 per year. We find that the localization costs of following up low-significance candidates are marginal, as the same changes to FAR only increase distance/area localizations by less than a factor of 2 and increase volume localization by less than a factor of 4. We argue that EM follow-up thresholds for low-significance candidates should be set on the basis of alert purity ( P astro ) and not FAR. Ideally, such estimates of P astro would be provided by LIGO-Virgo, but in their absence we provide estimates of the average purity of the GW candidate alerts issued by LIGO-Virgo as a function of FAR for various LIGO-Virgo observing epochs.