Some data center networks have already started to use optical circuit switching (OCS) with potential performance benefits, including high capacity, low latency, and energy efficiency. This paper addresses a switching network design to maximize the network radix, i.e., the number of terminals connected to the network under the condition that a specified number of identical switches with the size N×N and the maximum admissible blocking probability are given. Previous work presented a two-stage twisted and folded Clos network (TF-Clos) with a blocking probability guarantee for OCS, which has a larger network radix than TF-Clos with a strict-sense non-blocking condition. Expanding the number of stages allows for enhancing the network radix. This paper proposes a model designing an OCS three-stage TF-Clos structure with a blocking probability guarantee to increase the network radix compared to the two-stage TF-Clos. We formulate the problem of obtaining the network configuration that maximizes the network radix as an optimization problem. We conduct an algorithm based on an exhaustive search to obtain a feasible solution satisfying the constraints of the optimization problem. This algorithm identifies the structure with the largest network radix in non-increasing order to avoid unnecessary searches. Numerical results show that the proposed model achieves a larger network radix than the two-stage model.