The phase transition sequence of a PbHf0.7Sn0.3O3 single crystal was investigated by using Brillouin spectroscopy. The longitudinal acoustic mode showed three distinct changes at similar to 370 K, similar to 455 K and similar to 495 K which corresponded to the antiferroelectric 1 -> antiferroelectric 2 -> intermediate -> paraelectric phase transitions upon heating. The paraelectric phase was characterized by softening acoustic mode, increasing acoustic damping and the appearance of quasielastic central peaks which were observed upon approaching the phase transition temperature. The relaxation times derived from the acoustic mode anomalies and the central peak were consistent with each other and showed a slowing-down behavior. These precursor phenomena were attributed to the polar clusters having broken inversion symmetry. The substitution of Hf with Sn induced a very soft intermediate phase between the paraelectric and the antiferroelectric phases, where the longitudinal acoustic mode exhibited the lowest mode frequency along with substantially high acoustic damping. The transverse acoustic mode, which was observed in the two antiferroelectric phases, did not appear in the intermediate phase. These results indicated that the average symmetry of the intermediate phase may be cubic consisting of fine ferroelastic domains.