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The physics of the Mott metal-insulator transition (MIT) has attracted huge interest in the last decades. However, despite broad efforts, some key theoretical predictions are still lacking experimental confirmation. In particular, it is not clear whether the large coherent Fermi surface survives in immediate proximity to the bandwidth-controlled first-order MIT. A quantitative experimental verification of the predicted behavior of the quasiparticle effective mass, renormalized by many-body interactions, is also missing. Here we address these issues by employing organic $κ$-type salts as exemplary quasi-two-dimensional bandwidth-controlled Mott insulators and gaining direct access to their charge carrier properties via magnetic quantum oscillations. We trace the evolution of the effective cyclotron mass as the conduction bandwidth is tuned very close to the MIT by means of precisely controlled external pressure. We find that the sensitivity of the mass renormalization to tiny changes of the bandwidth is significantly stronger than theoretically predicted and is even further enhanced upon entering the transition region where the metallic and insulating phases coexist. On the other hand, even at its very edge of stability the metallic ground state preserves a large coherent Fermi surface with no significant enhancement of scattering.