Modern navigation systems do not only provide shortest paths but also some alternative paths to provide more options to the users. This paper is the first to study the problem of continuously reporting alternative paths for a user traveling along a given path. Specifically, given a path P on which a user is traveling, we continuously report to the user k paths from the user's current location on P to the target t. We present several algorithms each improving on the previous based on non-trivial observations and novel optimisations. The proposed algorithms maintain and exploit the previously computed useful information to efficiently update the k alternative paths as the user moves. We provide space and time complexity analysis for each proposed algorithm. We conduct a comprehensive experimental study on large real-world road networks. The results demonstrate that the proposed algorithms are up to several orders of magnitude faster than the straightforward algorithms.