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In this work, the concept of a multipurpose mission that can explore both the Moon and Mars with a single launch is proposed, and potential launch opportunities are analyzed to establish an early-phase trajectory concept. The proposed mission applies the concept of a piggyback ride to a small-sized lunar probe, i.e., the daughtership, of the main Mars orbiter, i.e., the mothership. For the trajectory design, the Earth-Moon-Mars gravity assist (EMMGA) trajectory is adopted for the mothership to reach Mars, and the daughtership is assumed to be released from the mothership during lunar flyby. To investigate the early-phase feasibility of the proposed mission, the launch windows have been analyzed and the associated delta-Vs have been directly compared with the solutions obtained for typical Earth-Mars direct (EMD) transfer options. The identified launch windows (in the years 2031 and 2045) could be the strongest candidates for the proposed conceptual mission. Under the current assumptions, up to approximately 15% (in 2031) and 9% (in 2045), more dry mass is expected to be delivered to Mars by appropriately selecting one of the currently available launch vehicles, regardless of whether the EMMGA transfer option is used. For missions around the Moon using a SmallSat in 2031, the feasibility of a lunar orbiter case and an impactor case is briefly analyzed based on the delta-Vs required to divert the SmallSat from the mothership. Although the current work is performed under numerous assumptions for a simplified problem, the narrowed candidate launch window from the current work represents a good starting point for more detailed trajectory design optimization and analysis to realize the proposed conceptual mission. [ABSTRACT FROM AUTHOR]