Multi-stage semi-open queuing networks (SOQNs) are widely used to analyze the performance of multi-stage manufacturing systems and automated warehousing systems. While there are several methods available for solving single-stage SOQNs, solution methods for multi-stage SOQNs are limited. Decomposition of a multi-stage SOQN into single-stage SOQNs and evaluation of an individual single-stage SOQN is a possibility. However, the challenge lies in obtaining the job departure process information from an upstream single-stage SOQN to evaluate the performance of a downstream single-stage SOQN. In this paper, we propose a two-moment approximation approach for estimating the squared coefficient of variation of the job inter-departure time from a single-stage SOQN, which can serve as an input to link multi-stage SOQNs. Using numerical experiments, we test the robustness of the proposed approach for various input parameter settings for both single and multi-class jobs. We find that the proposed approach works quite well, particularly when the coefficient of variation of the job inter-arrival time is less than two. We demonstrate the efficacy of the proposed approach using a case study on a multi-tier shuttle-based compact storage system and benchmark our results with an existing approach. The results indicate that our approach yields more accurate estimates of the performance measures in comparison to the existing approach in the literature.