With progress toward a practical quantum computer has come an increasingly rapid search for quantum-safe, secure communication schemes that do not rely on discrete logarithm or factorization problems. One such encryption scheme, Multi-path Switching with Secret Sharing (MSSS), combines secret sharing with multi-path switching to achieve security as long as the adversary does not have global observability of all paths and thus cannot capture enough shares to reconstruct messages. MSSS assumes that sending a share on a path is an atomic operation and all paths have the same delay. In this paper, we identify a side-channel vulnerability for MSSS, created by the fact that in real networks, sending a share is not an atomic operation as paths have multiple hops and different delays. This channel, referred to as Network Data Remanence (NDR), is present in all schemes like MSSS whose security relies on transfer atomicity and all paths having same delay. We demonstrate the presence of NDR in a physical testbed. We then identify two new attacks that aim to exploit the side channel, referred to as NDR Blind and NDR Planned, propose an analytical model to analyze the attacks, and demonstrate them using an implementation of MSSS based on the ONOS SDN controller. Finally, we present a countermeasure for the attacks and show its effectiveness in simulations and Mininet experiments.