  We reveal a hierarchical, multilayer organization of finite components --
i.e., tendrils and tubes -- around the giant connected components in directed
networks and propose efficient algorithms allowing one to uncover the entire
organization of key real-world directed networks, such as the World Wide Web,
the neural network of \emph{Caenorhabditis elegans}, and others. With
increasing damage, the giant components decrease in size while the number and
size of tendril layers increase, enhancing the susceptibility of the networks
to damage.
