We show that the smoothed complexity of the FLIP algorithm for local Max-Cut is at most $\smash{φn^{O(\sqrt{\log n})}}$, where $n$ is the number of nodes in the graph and $φ$ is a parameter that measures the magnitude of perturbations applied on its edge weights. This improves the previously best upper bound of $φn^{O(\log n)}$ by Etscheid and Röglin. Our result is based on an analysis of long sequences of flips, which shows~that~it is very unlikely for every flip in a long sequence to incur a positive but small improvement in the cut weight. We also extend the same upper bound on the smoothed complexity of FLIP to all binary Maximum Constraint Satisfaction Problems.