Launch mission planning and ascent guidance is one of the most notable engineering fields where optimization tools and optimal control theory have found routine applications. Optimality is critical to achieve the full performance of a launch vehicle. In the case of a multi-stage launch, allowing for optimized coast arcs between burns can significantly reduce propellant consumption and enhance mission capability. These coast arcs, however, render the optimal control problem more sensitive and increase algorithm convergence difficulties. This work presents detailed improvements to an analytical multiple-shooting (AMS) method for reliable generation of the optimal exo-atmospheric ascent trajectory. The trajectory consists of two burns separated by an optimized coast arc. The problem is in closed-form and quadratures. A strong effort is made in increasing the robustness, reliability, and flexibility of the algorithm. The improvements include an introduction of a more sophisticated numerical method, replacement of the current coast arc solution with a completely general, compact, and easily implementable method capable of determining the solution to machine precision, and a direct treatment of the orbital insertion conditions and resulting unknown multipliers. An aerospace industry standard trajectory optimization software, Optimal Trajectories by Implicit Simulation (OTIS), is employed to compare the results and verify the improved AMS algorithm. A wide range of mission scenarios are tested using the algorithm in open-loop solution and closed-loop simulation.