Resonant optical excitation of certain molecular vibrations in κ-(BEDT-TTF)_{2}Cu[N(CN)_{2}]Br has been shown to induce transient superconductinglike optical properties at temperatures far above equilibrium T_{c}. Here, we report experiments across the bandwidth-tuned phase diagram of this class of materials, and study the Mott insulator κ-(BEDT-TTF)_{2}Cu[N(CN)_{2}]Cl and the metallic compound κ-(BEDT-TTF)_{2}Cu(NCS)_{2}. We find nonequilibrium photoinduced superconductivity only in κ-(BEDT-TTF)_{2}Cu[N(CN)_{2}]Br, indicating that the proximity to the Mott insulating phase and possibly the presence of preexisting superconducting fluctuations are prerequisites for this effect.