We carry out a search for strong-lens systems containing high-redshift lens galaxies with the goal of extending strong-lensing-assisted galaxy evolutionary studies to earlier cosmic time. Two strong-lens classifiers are constructed from a deep residual network and trained with datasets of different lens-redshift and brightness distributions. We classify a sample of 5356628 pre-selected objects from the Wide-layer fields in the second public data release of the Hyper Suprime-Cam Subaru Strategic Program (HSC-SSP) by applying the two classifiers to their HSC gri-filter cutouts. Cutting off at thresholds that correspond to a false-positive rate of 10^-3^ on our test set, the two classifiers identify 5,468 and 6,119 strong-lens candidates. Visually inspecting the cutouts of those candidates results in 735 grade-A or B strong-lens candidates in total, of which 277 candidates are discovered for the first time. This is the single largest set of galaxy-scale strong-lens candidates discovered with HSC data to date, and nearly half of it (331/735) contains lens galaxies with photometric redshifts above 0.6. Our discoveries will serve as a valuable target list for ongoing and scheduled spectroscopic surveys such as the Dark Energy Spectroscopic Instrument, the Subaru Prime Focus Spectrograph project, and the Maunakea Spectroscopic Explorer.