We present a search for binary neutron star (BNS) mergers that produced gravitational waves during the first observing run of the Advanced Laser Interferometer Gravitational-Wave Observatory (LIGO), and gamma-ray emission seen by either the Swift-Burst Alert Telescope (BAT) or the Fermi-Gamma-ray Burst Monitor (GBM), similar to GW170817 and GRB 170817A. We introduce a new method using a combined ranking statistic to detect sources that do not produce significant gravitational-wave or gamma-ray burst candidates individually. The current version of this search can increase by 70% the detections of joint gravitational-wave and gamma-ray signals. We find one possible candidate observed by LIGO and Fermi-GBM, 1-OGC 151030, at a false alarm rate of 1 in 13 yr. If astrophysical, this candidate would correspond to a merger at Mpc with source-frame chirp mass of . If we assume that the viewing angle must be <30° to be observed by Fermi-GBM, our estimate of the distance would become Mpc. By comparing the rate of BNS mergers to our search-estimated rate of false alarms, we estimate that there is a 1 in 4 chance that this candidate is astrophysical in origin.