DESTINY+ is a medium-class interplanetary mission, selected by the Japan Aerospace Exploration Agency for potential launch windows in the first half of 2020s. The mission will demonstrate innovative spacecraft subsystem technologies, including a new type of ion engine for future missions. The mission will also collect scientific data through high-speed flyby observations and dust measurements from asteroid (3200) Phaethon and its related body (155140) 2005 UD, to understand their origin and reveal the content of extraterrestrial dust in the context of origin of life. The limited control authority on the spacecraft, the orbits of the target asteroids, and the specific mission requirements pose a challenging task for the trajectory design of DESTINY+. Multiple-target low-thrust optimal trajectories are explored in this paper to fulfill the goals of the DESTINY+ mission. An effective methodology is presented to convert feasible impulsive transfer solutions into low-thrust initial guesses and combine with gravity-assist maneuvers to reveal new high-fidelity optimal trajectories in real ephemeris models. The early mission analysis results demonstrate multitudes of flyby opportunities that provide robustness against programmatic and operational delays in the mission schedule.@en