The sky-averaged cosmological 21 cm signal can improve our understanding of the evolution of the early Universe from the Dark Age to the end of the Epoch of Reionization. Although the EDGES experiment reported an absorption profile of this signal, there have been concerns about the plausibility of these results, motivating independent validation experiments. One of these initiatives is the Mapper of the IGM Spin Temperature (MIST), which is planned to be deployed at different remote locations around the world. One of its key features is that it seeks to comprehensively compensate for systematic uncertainties through detailed modeling and characterization of its different instrumental subsystems, particularly its antenna. Here we propose a novel optimizing scheme which can be used to design an antenna applied to MIST, improving bandwidth, return loss, and beam chromaticity. This new procedure combines the Particle Swarm Optimization (PSO) algorithm with a commercial electromagnetic simulation software (HFSS). We improved the performance of two antenna models: a rectangular blade antenna, similar to the one used in the EDGES experiment, and a trapezoidal bow-tie antenna. Although the performance of both antennas improved after applying our optimization method, we found that our bow-tie model outperforms the blade antenna by achieving lower reflection losses and beam chromaticity in the entire band of interest. To further validate the optimization process, we also built and characterized 1:20 scale models of both antenna types showing an excellent agreement with our simulations.