This paper studies the integration of machine-learned advice in overlay networks in order to adapt their topology to the incoming demand. Such demand-aware systems have recently received much attention, for example in the context of data structures (Fu et al. in ICLR 2025, Zeynali et al. in ICML 2024). We in this paper extend this vision to overlay networks where requests are not to individual keys in a data structure but occur between communication pairs, and where algorithms have to be distributed. In this setting, we present an algorithm that adapts the topology (and the routing paths) of the overlay network to minimize the hop distance travelled by bit, that is, distance times demand. In a distributed manner, each node receives an (untrusted) prediction of the future demand to help him choose its set of neighbors and its forwarding table. This paper focuses on optimizing the well-known skip list networks (SLNs) for their simplicity. We start by introducing continuous skip list networks (C-SLNs) which are a generalization of SLNs specifically designed to tolerate predictive errors. We then present our learning-augmented algorithm, called LASLiN, and prove that its performance is (i) similar to the best possible SLN in case of good predictions ($O(1)$-consistency) and (ii) at most a logarithmic factor away from a standard overlay network in case of arbitrarily wrong predictions ($O(\log^2 n)$-robustness, where $n$ is the number of nodes in the network). Finally, we demonstrate the resilience of LASLiN against predictive errors (ie, its smoothness) using various error types on both synthetic and real demands.