Several systems rely on traceroute to track a large number of Internet paths as they change over time. Monitoring systems perform this task by remapping paths periodically or whenever a change is detected. This paper shows that such complete remapping is inefficient, because most path changes are localized to a few hops of a path. We develop RemapRoute, a tool to remap a path locally given the previously known path and a change point. RemapRoute sends targeted probes to locate and remap the often few hops that have changed. Our evaluation with trace-driven simulations and in a real deployment shows that local remapping reduces the average number of probes issued during remapping by 63% and 79%, respectively, when compared with complete remapping. At the same time, our results show that local remapping has little impact on the accuracy of inferred paths.