The Mott metal-insulator transition-a drastic manifestations of Coulomb interactions among electrons-is the first-order transition of clear discontinuity, as shown by various experiments and the celebrated dynamical mean-field theory. Recent theoretical works, however, suggest that the transition is continuous if the Mott insulator carries an exotic spin liquid with a spinon Fermi surface. Here, we demonstrate the case of a quasi-continuous Mott transition from a Fermi liquid to a spin liquid in an organic triangular-lattice system k-(ET)2Cu2(CN)3. Transport experiments performed under fine pressure tuning find that, as the Mott transition is approached, the Fermi-liquid coherence temperature continuously falls to the scale of kelvins with divergent quasi-particle decay rate in the metal side and the charge gap gradually closes in the insulator side. The Clausius-Clapeyron analysis of the pressure-temperature phase diagram provides thermodynamic evidence for the extremely weak first-order nature of the Mott transition. These results suggest that the spin liquid hosts a spinon Fermi surface, which turns into an electron Fermi surface when charges are Mott delocalized.