As privacy-preserving becomes a pivotal aspect of deep learning (DL) development, multi-party computation (MPC) has gained prominence for its efficiency and strong security. However, the practice of current MPC frameworks is limited, especially when dealing with large neural networks, exemplified by the prolonged execution time of 25.8 seconds for secure inference on ResNet-152. The primary challenge lies in the reliance of current MPC approaches on additive secret sharing, which incurs significant communication overhead with non-linear operations such as comparisons. Furthermore, additive sharing suffers from poor scalability on party size. In contrast, the evolving landscape of MPC necessitates accommodating a larger number of compute parties and ensuring robust performance against malicious activities or computational failures. In light of these challenges, we propose SSNet, which for the first time, employs Shamir's secret sharing (SSS) as the backbone of MPC-based ML framework. We meticulously develop all framework primitives and operations for secure DL models tailored to seamlessly integrate with the SSS scheme. SSNet demonstrates the ability to scale up party numbers straightforwardly and embeds strategies to authenticate the computation correctness without incurring significant performance overhead. Additionally, SSNet introduces masking strategies designed to reduce communication overhead associated with non-linear operations. We conduct comprehensive experimental evaluations on commercial cloud computing infrastructure from Amazon AWS, as well as across diverse prevalent DNN models and datasets. SSNet demonstrates a substantial performance boost, achieving speed-ups ranging from 3x to 14x compared to SOTA MPC frameworks. Moreover, SSNet also represents the first framework that is evaluated on a five-party computation setup, in the context of secure DL inference.