The integration of automation and data-driven methodologies offers a promising approach to accelerating materials discovery in energy storage research. Thus far, in battery research, coin-cell assembly has advanced to become nearly fully automated but remains largely disconnected from data-driven methods. To bridge the disconnect, this work presents a self-driving laboratory framework to accelerate electrolyte discovery by integrating automated coin-cell assembly, galvanostatic cycling of LiFePO₄||Li₄Ti₅O₁₂ organic-aqueous full cells, and Bayesian optimization for selecting subsequent experiments based on prior results. The study explored an organic-aqueous hybrid electrolyte system comprising four co-solvents and two lithium-conducting salts. Using this framework, cells with an optimized electrolyte cycled with at least 94% Coulombic efficiency. Additionally, online electrochemical mass spectrometry revealed that the optimized organic co-solvents successfully mitigated the parasitic hydrogen evolution reaction. The results highlight the potential of combining Bayesian optimization with autonomous full-cell experimentation while contributing new electrolyte design insights for next-generation aqueous batteries.