Quantum spin liquid is a nontrivial magnetic state of longstanding interest, in which spins are strongly correlated and entangled but do not order; further intriguing is its doped version, which possibly hosts strange metal and unconventional superconductivity. A promising candidate of the doped spin liquid is a triangular-lattice organic conductor, κ-(BEDT-TTF)4Hg2.89Br8, recently found to hold metallicity, spin-liquid-like magnetism, and BEC-like superconductivity. The nature of the metallic state with the spin-liquid behaviour is awaiting to be further clarified. Here, we report the thermoelectric signature that mobile holes in the spin liquid background are in a quantum critical state and it pertains to the BEC-like superconductivity. The Seebeck coefficient divided by temperature,S/T, is enhanced on cooling with logarithmic divergence indicative of quantum criticality. Furthermore, the logarithmic enhancement is correlated with the superconducting transition temperature under pressure variation, and the temperature and magnetic field profile ofS/Tupon the superconducting transition change with pressure in a consistent way with the previously suggested BEC-BCS crossover. The present results reveal that the quantum criticality in a doped spin liquid emerges in a phase, not at a point, and is involved in the unconventional BEC-like nature.