The quickest contraflow in a single-source-single-sink network is a dynamic flow that minimizes the time horizon of a given flow value at the source to be sent to the sink allowing arc reversals. Because of the arc reversals, for a sufficiently large value of the flow, the residual capacity of all or most of the paths towards the source, from a given node, may be zero or reduced significantly. In some cases, e.g., for the movement of facilities to support the evacuation in an emergency, it is imperative to save a path from a given node towards the source. We formulate such a problem as a bi-criteria optimization problem, in which one objective minimizes the length of the path to be saved from a specific node towards the source, and the other minimizes the quickest time of the flow from the source towards the sink allowing arc reversals. We propose an algorithm based on the epsilon-constraint approach to find non-dominated solutions.