For linear systems with a time-varying input delay, the predictor feedback controller and exponential stability have been established. However, the now-classical approach of representing the delay by a transport partial differential equation (PDE) on a strictly positive and constant spatial domain precludes the possibility of the delay assuming the zero value at any time instant. To eliminate this limitation, we provide a new representation of the delay by a transport equation with a time-varying spatial domain. The resulting backstepping approach leads to the same predictor feedback that was previously designed by the last author. However, the controller derivation and the stability analysis are quite different, even though both the controller and the assumptions are the same. A representative example is provided to illustrate the methodology and results.