The time-gap regulation problem for a cascaded system consisting of platooned automated vehicles following a leading non-automated vehicle is investigated in this article. Under the assumption of uniform boundedness of the acceleration of the leading vehicle, a control design scheme is proposed via an extension of integral backstepping control method, where additional terms that counter the impact due to the speed change of the non-automated vehicle are used. Each automated vehicle is actuated by one backstepping controller, demonstrated by a recursive control design procedure based on induction. As a result, both the time-gap error and the speed error between each pair of consecutive vehicles are proven to be ultimately bounded by some constants that can be tuned to be arbitrarily close to zero. In particular, the regulated time-gap guarantees enough time for the following vehicle in each pair to react to the velocity change of its preceding vehicle. Simulation is carried out to validate the proposed controllers.