Stochastic block models learn group structures between nodes sharing similar connectivity patterns. Recent developments have extended this approach to multiple connected networks, often within multilayer or multiplex architectures. However, most formulations still rely on the strong assumption that all networks share a single node partition. In many applications, this assumption is too restrictive: connected networks may exhibit distinct but hierarchically dependent clustering structures. For example, in criminal networks that track the number of meetings attended by criminals at different levels of the criminal organization, the clustering of nodes at one level may naturally fragment into more detailed communities at another level, reflecting different but related organizational principles. To address this, we introduce partially exchangeable enriched stochastic block models, a new class of Bayesian network models that jointly capture multiple layers of dependency through partially exchangeable priors on node partitions. Building on the partially exchangeable stochastic block model of Durante et al (2025), we extend its construction to an enriched framework where two partitions are linked by a nested structure. The first partition, which governs a node-colored network, is informed by the covariates of the nodes, while the second is generated by fragmenting this coarser structure into nested subgroups. This joint prior is derived from an enriched Gibbs-type process, ensuring partial exchangeability while flexibly adapting to shared and network-specific clustering behaviors. Inference is performed via a collapsed Gibbs sampler using a cut-posterior perspective to improve mixing. Preliminary results on simulated data and in a study of joint participation in summits within a complex mafia organization highlight the strengths of the proposed formulation and its ability to integrate and learn relevant structures in networks.