This study investigates whether reoptimization can help in solving the closest substring problem. We are dealing with the following reoptimization scenario. Suppose, we have an optimal l-length closest substring of a given set of sequences S. How can this information be beneficial in obtaining an (l+k)-length closest substring for S? In this study, we show that the problem is still computationally hard even with k=1. We present greedy approximation algorithms that make use of the given information and prove that it has an additive error that grows as the parameter k increases. Furthermore, we present hard instances for each algorithm to show that the computed approximation ratio is tight. We also show that we can slightly improve the running-time of the existing polynomial-time approximation scheme (PTAS) for the original problem through reoptimization.