Multipartite non-Gaussian entanglement constitutes a core resource for continuous-variable quantum networks. Although three-mode spontaneous parametric down-conversion (SPDC)–a process capable of deterministically generating non-Gaussian entanglement–has recently been observed in superconducting circuits, the weak nonlinearity inherent to such systems limits the practical utility of the generated non-Gaussian entanglement. In this work, we propose an entanglement-enhancing protocol based on a cascaded nonlinear model, where three-mode SPDC is sequentially followed by two-mode SPDC. We adopt quantum relative entropy and logarithmic negativity as independent quantifiers to characterize how the coupling strength of two-mode SPDC shapes the non-Gaussianity and entanglement of the system, respectively. Our results show that two-mode SPDC does not enhance the overall non-Gaussianity of the full system; nevertheless, it can boost global entanglement and amplify the non-Gaussianity of specific bipartite subsystems. Furthermore, we analyze the correlations among the three output modes and reveal the entanglement of these output modes in high-order moments by employing the positive partial transpose criterion based on the high-order covariance matrix. Our results provide a theoretical foundation for the manipulation of non-Gaussian entanglement via cascaded nonlinearity, which is expected to facilitate the early application of non-Gaussian entanglement in quantum information protocols.