We conduct a search for galaxy-scale strong gravitational lens systems in Data Release 4 of the Hyper Suprime-Cam Subaru Strategic Program (HSC SSP), consisting of data taken up to the S21A semester. We select 103191 luminous red galaxies from the Baryon Oscillation Spectroscopic Survey (BOSS) sample that have deep multiband imaging from the HSC SSP and use the YATTALENS algorithm to identify lens candidates with blue arc-like features automatically. The candidates are visually inspected and graded based on their likelihood of being a lens. We find eight definite lenses, 28 probable lenses, and 138 possible lenses. The new lens candidates generally have lens redshifts in the range 0.3<~z_L_<~0.9, a key intermediate redshift range to study the evolution of galaxy structure. Follow-up spectroscopy will confirm these new lenses and measure source redshifts to enable detailed lens modeling.