Optical circuit switching provides high-capacity and low-latency data transmission capabilities. A Clos network is well known as a class of non-blocking switching networks. A prior study addressed designing a three-stage twisted and folded Clos network (3TF), which has a larger switching network size, i.e., the number of terminals connected to the network, than its two-stage variants. That study used a sufficient condition for strict-sense non-blocking (SNB) in 3TF to solve an optimization design problem. However, this condition may be stricter than necessary to guarantee strict-sense non-blocking, suggesting that by using the necessary and sufficient condition, the constraints can likely be relaxed, allowing for configurations with a larger switching network size. No study has addressed the necessary and sufficient condition for SNB in the whole 3TF. A past study derived the SNB condition for the three-stage folded Clos (3F), the basis of 3TF, assuming that connections within the same first- and second-stage group loop back no further than the second stage. Its necessary and sufficient condition in the general case, however, remains unexamined, despite an initial study on the SNB condition for 3F. This paper proposes a design model of 3TF based on the necessary and sufficient condition for strict-sense non-blocking. We derive and prove the necessary and sufficient condition for strict-sense non-blocking in the whole 3TF. The proposed design model considers two cases: one where no blocking is allowed and another where the blocking probability does not exceed a given admissible threshold, based on our derived necessary and sufficient condition for strict-sense non-blocking in 3TF. We formulate the problem based on the necessary and sufficient condition to obtain the configuration of the proposed model that maximizes the switching network size. We theoretically show that the maximum switching network size obtained by solving the proposed model is always larger than or equal to that obtained by solving the existing 3TF design model. Numerical results show that the proposed model achieves a larger switching network size than the existing 3TF design model.