The tropical variety of a prime polynomial ideal is a pure, connected polyhedral fan, that may be computed by traversal. To start this traversal a single cone is required. So far no good method for finding a starting cone was known. In this talk we propose to find one by stably intersecting with coordinate hyperplanes until a tropical curve is obtained. The key idea is that doing stable intersections translates into doing Groebner basis computations over a field of rational functions. Using a recursive reformulation of Chan's tropical curve algorithm, finding a single ray in a curve is often easy. The tropical cone is then obtained by repeatedly finding such rays according to an already known heuristic strategy for tropical cone construction.