More sustainable chemical processes require the selection of suitable molecules, which can be supported by computer-aided molecular design (CAMD). CAMD often generates and evaluates molecular structures using genetic algorithms. However, genetic algorithms can suffer from slow convergence, and might yield suboptimal solutions. In response to these challenges, this work presents a method to fine-tune a genetic algorithm for CAMD. The proposed method builds on the COSMO-CAMD framework that utilizes a genetic algorithm for solving optimization-based molecular design problems and COSMO-RS for predicting physical properties of molecules. The key idea of the proposed method is to integrate results from a fast large-scale molecular screening into the molecular design framework through an automated fragmentation procedure. By generating a promising initial population and constructing a tailored fragment library, our method enables a targeted initialization of the genetic algorithm, referred to as warm-start. The proposed method is applied in two case studies to design solvents for extracting γ-valerolactone and phenol, respectively, from aqueous solutions. Compared to the benchmark method, the warm-started COSMO-CAMD framework achieves a 70% faster convergence, discovers 4-fold more top-performing candidate molecules, and identifies seven tailored molecular fragments, culminating in the discovery of two novel solvents specifically for the phenol case. The optimal solvent is found in all computational runs. Overall, the warm-started COSMO-CAMD framework significantly improves efficiency, effectiveness, and robustness of molecular design.