Quantum routers direct a quantum signal from one input path to a quantum superposition of multiple output paths and are considered important elements of future quantum networks. To extend the scalability of quantum networks, multi-layer quantum routers, which allow for further superposition of paths, were proposed. Due to the numerous limitations of current quantum devices, quantum error mitigation methods become potential solutions for realizing practical quantum routers in the near term. Zero-Noise Extrapolation (ZNE) and Clifford Data Regression (CDR) are two promising quantum error mitigation methods. Based on the characteristics of these two methods, we propose a new method, named extrapolated CDR (eCDR). We benchmark the performance of multi-layer quantum routers implemented on current superconducting quantum devices instantiated with the ZNE, CDR, and eCDR methods. Our experimental results show that the new eCDR method improves the fidelity result of the two-layer quantum router. Our work highlights how new mitigation methods built from different components of pre-existing methods, and designed with a core application in mind, can lead to significant performance enhancements.