  We report the discovery by the HATSouth project of 5 new transiting hot
Jupiters (HATS-54b through HATS-58Ab). HATS-54b, HATS-55b and HATS-58Ab are
prototypical short period ($P = 2.5-4.2$ days, $R_p\sim1.1-1.2$ $R_J$)
hot-Jupiters that span effective temperatures from 1350 K to 1750 K, putting
them in the proposed region of maximum radius inflation efficiency. The HATS-58
system is composed of two stars, HATS-58A and HATS-58B, which are detected
thanks to Gaia DR2 data and which we account for in the joint modelling of the
available data --- with this, we are led to conclude that the hot jupiter
orbits the brighter HATS-58A star. HATS-57b is a short-period (2.35-day)
massive (3.15 $M_J$) 1.14 $R_J$, dense ($2.65\pm0.21$ g cm$^{-3}$) hot-Jupiter,
orbiting a very active star ($2\%$ peak-to-peak flux variability). Finally,
HATS-56b is a short period (4.32-day) highly inflated hot-Jupiter (1.7 $R_J$,
0.6 $M_J$), which is an excellent target for future atmospheric follow-up,
especially considering the relatively bright nature ($V=11.6$) of its F dwarf
host star. This latter exoplanet has another very interesting feature: the
radial velocities show a significant quadratic trend. If we interpret this
quadratic trend as arising from the pull of an additional planet in the system,
we obtain a period of $P_c = 815^{+253}_{-143}$ days for the possible planet
HATS-56c, and a minimum mass of $M_c\sin i_c = 5.11 \pm 0.94$ $M_J$. The
candidate planet HATS-56c would have a zero-albedo equilibrium temperature of
$T_\textrm{eq}=332\pm 50$ K, and thus would be orbiting close to the habitable
zone of HATS-56. Further radial-velocity follow-up, especially over the next
two years, is needed to confirm the nature of HATS-56c.
