We introduce the concept of local controllability for a control system of ordinary differential equations with free time and provide sufficient conditions for its controllability. As a direct consequence, for time-optimal problems, we obtain necessary conditions for a local infimum-this notion generalizes that of an optimal trajectory. These conditions constitute a family of relations, each of which has the form of a maximum principle. Examples are given to demonstrate the meaningfulness of the necessary conditions obtained in the present paper, which generalize and strengthen the Pontryagin maximum principle.