  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) $\sim 300$~deg$^2$ 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$\sigma$) 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$^{+24}_{-11}$~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 with
a half-light radius of $r_h = 90^{+42}_{-17}$~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 $M_V = -0.33^{+0.75}_{-0.87}$~mag and $r_h =
47^{+19}_{-13}$~pc. Using simulations of $\Lambda$-dominated cold dark matter
models, we predict that we should find one or two new MW satellites from $\sim
300$~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$^2$ will
provide robust insights into the missing satellites problem.
