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 T_s1 ​= ​265 ​K and T_s2 ​= ​123 ​K. At T_s1, the magnetic susceptibility is enormously reduced and a structural transition to a monoclinic structure takes place, while relatively small changes are observed at T_s2. 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.