<p>This paper proposes a novel Multi-Order Single Shooting (MOSS) scheme that serves to expand the domain of convergence of indirect optimization methods, thereby alleviating the burden of requiring good initial guesses. This is achieved by incorporating the information of higher-order sensitivities into the update step of the shooting method. The necessary higher-order sensitivities are computed using a derivative-free approach that reduces the computational costs associated with traditional sensitivity computation methods. In this paper, MOSS is successfully applied in designing low-thrust minimum-fuel pathways in the Earth-Moon system. Significant improvement in convergence is observed when compared against a traditional shooting method. MOSS is also used to identify heteroclinic connections between planar periodic orbits as solutions to the minimum-fuel problem, without requiring traditional dynamical systems theory approaches.</p>