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We present the results from a search for new Milky Way (MW) satellites fromthe first two years of data from the Hyper Suprime-Cam (HSC) Subaru StrategicProgram (SSP) $\sim 300$~deg$^2$ and report the discovery of a highlycompelling ultra-faint dwarf galaxy candidate in Cetus. This is the secondultra-faint dwarf we have discovered after Virgo~I reported in our previouspaper. This satellite, Cetus~III, has been identified as a statisticallysignificant (10.7$\sigma$) spatial overdensity of star-like objects, which areselected from a relevant isochrone filter designed for a metal-poor and oldstellar population. This stellar system is located at a heliocentric distanceof 251$^{+24}_{-11}$~kpc with a most likely absolute magnitude of $M_V = -2.4\pm 0.6$~mag estimated from a Monte Carlo analysis. Cetus~III is extended witha half-light radius of $r_h = 90^{+42}_{-17}$~pc, suggesting that this is afaint dwarf satellite in the MW located beyond the detection limit of the SloanDigital Sky Survey. Further spectroscopic studies are needed to assess thenature of this stellar system. We also revisit and update the parameters forVirgo~I finding $M_V = -0.33^{+0.75}_{-0.87}$~mag and $r_h =47^{+19}_{-13}$~pc. Using simulations of $\Lambda$-dominated cold dark mattermodels, we predict that we should find one or two new MW satellites from $\sim300$~deg$^2$ HSC-SSP data, in rough agreement with the discovery rate so far.The further survey and completion of HSC-SSP over $\sim 1,400$~deg$^2$ willprovide robust insights into the missing satellites problem.

Searches for New Milky Way Satellites from the First Two Years of Data of the Subaru/Hyper Suprime-Cam Survey: Discovery of Cetus~III