For hybrid systems where the continuous state space is partitioned by switching manifolds, the discrete state changes autonomously if the continuous state hits a manifold. Recently, we introduced a version of the minimum principle for the optimal control of such systems, where necessary optimality conditions were provided for the case that a trajectory passes through the intersection of switching manifolds [1]. Further, we introduced an algorithm based on these conditions for computing optimal controls by varying not only the continuous state on switching manifolds based on gradient information, but also the sequence of discrete states [2]. It was shown that, the combinatorial complexity of former computational schemes based on the minimum principle can be avoided, since not all possible discrete state sequences need to be analyzed separately. In this note, theoretical aspects of the algorithm and practical considerations for a successful implementation are discussed in detail. Further, a comparison to existing algorithms for the solution of hybrid optimal control problems with autonomous switching is presented. The efficiency of the proposed algorithm is demonstrated by a novel numerical example.