Use the tool below to find a route that covers every path. First, execute a query against the Overpass API to fetch all tracks within a certain boundary, and then click on a point on the graph to select a starting point. To play around with the API and get help with the syntax (it is not terribly intuitive), visit overpass turbo. For some of the background on what motivated this tool, please read this post.
Enter your Overpass query and click "Fetch Tracks."