It is proved that for any hexagonal system, there is a peak and a valley which are border adjacent (the path from the peak to the valley along the border is monotone) and such that either the wetting border of the the peak contains no peaks or the catching border of the valley contains no valleys. Based on this property, a revised peeling algorithm is given to find a perfect matching for a hexagonal system if it exists. As a consequence, it is shown that the matching between peaks and valleys induced by perfect path systems is unique.