We have carried out a systematic search for galaxy-scale strong lenses in multiband imaging from the Hyper Suprime-Cam (HSC) survey. Our automated pipeline, based on realistic strong-lens simulations, deep neural network classification, and visual inspection, is aimed at efficiently selecting systems with wide image separations (Einstein radii ~1.0-3.0"), intermediate redshift lenses (z~0.4-0.7), and bright arcs for galaxy evolution and cosmology. We classified gri images of all 62.5 million galaxies in HSC Wide with i-band Kron radius >0.8" to avoid strict preselections and to prepare for the upcoming era of deep, wide-scale imaging surveys with Euclid and Rubin Observatory. We obtained 206 newly-discovered candidates classified as definite or probable lenses with either spatially-resolved multiple images or extended, distorted arcs. In addition, we found 88 high-quality candidates that were assigned lower confidence in previous HSC searches, and we recovered 173 known systems in the literature. These results demonstrate that, aided by limited human input, deep learning pipelines with false positive rates as low as ~0.01% can be very powerful tools for identifying the rare strong lenses from large catalogs, and can also largely extend the samples found by traditional algorithms. We provide a ranked list of candidates for future spectroscopic confirmation.