We study the lead rhenium oxide PbRe2O6 as a candidate spin-orbit-coupled metal (SOCM), which has attracted much attention as a testing ground for studying unconventional Fermi liquid instability associated with a large spin-orbit interaction. The compound comprises a stack of modulated honeycomb lattices made of Re5+ (5d2) ions in a centrosymmetric R-3m structure at room temperature. Resistivity, magnetic susceptibility, and heat capacity measurements using single crystals reveal two successive first-order phase transitions at Ts1 = 265 K and Ts2 = 123 K. At Ts1, the magnetic susceptibility is enormously reduced and a structural transition to a monoclinic structure takes place, while relatively small changes are observed at Ts2. Surprisingly, PbRe2O6 bears a close resemblance to another SOCM candidate Cd2Re2O7 despite crucial differences in the crystal structure and probably in the electronic structure, suggesting that PbRe2O6 is an SOCM.