Hume can be overcome only on his own ground, or, more precisely, the ground on which he was trying to carry out his own considerations but which methodologically he himself was unable to secure sufficiently. He started out with nature as it presents itself to the eyes of the naive contemplator. In this nature there's one causative linkage, one necessary sequence of happening. He wanted to investigate consciousness of this linkage: what kind of consciousness it is and whether it is rational. All that kept him from finding the evident coherence that he sought was a half-baked theory of the nature of consciousness and especially of experience. It misled him on the conclusion as well, to explain away the phenomena from which he started out and without which his whole way of setting up the issue would become incomprehensible.
Edith Stein, "Sentient Causality," in Philosophy of Psychology and the Humanities, Baseheart and Sawicki, trs., Collected Works vol. 7, ICS Publications (Washington, D.C.: 2000), p. 4.