We report the discovery by the HATSouth project of five 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 similar to 1.1-1.2 R-J) hot Jupiters that span effective temperatures from 1350 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 modeling 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 +/- 0.21 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(-143)(+253) days for the possible planet HATS-56c, and a minimum mass of M-c sin i(c) = 5.11 +/- 0.94 M-J. The candidate planet HATS-56c would have a zero-albedo equilibrium temperature of T-eq = 332 +/- 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.