We present the results from a search for new Milky Way (MW) satellites from the first two years of data from the Hyper Suprime-Cam (HSC) Subaru Strategic Program (SSP) ∼300 deg2 and report the discovery of a highly compelling ultra-faint dwarf galaxy candidate in Cetus. This is the second ultra-faint dwarf we have discovered after Virgo I reported in our previous paper. This satellite, Cetus III, has been identified as a statistically significant (10.7 σ) spatial overdensity of star-like objects, which are selected from a relevant isochrone filter designed for a metal-poor and old stellar population. This stellar system is located at a heliocentric distance of 251-11+24 kpc with a most likely absolute magnitude of MV = -2.4 ± 0.6mag estimated from a Monte Carlo analysis. Cetus III is extended with a half-light radius of rh = 90-17+42 pc, suggesting that this is a faint dwarf satellite in the MW located beyond the detection limit of the Sloan Digital Sky Survey. Further spectroscopic studies are needed to assess the nature of this stellar system. We also revisit and update the parameters for Virgo I, finding MV = -0.33-0.87+0.75 mag and rh = 47-13+19 pc. Using simulations of Σ-dominated cold dark matter models, we predict that we should find one or two new MW satellites from ∼300 deg2 HSC-SSP data, in rough agreement with the discovery rate so far. The further survey and completion of HSC-SSP over ∼1400 deg2 will provide robust insights into the missing satellites problem.