As the importance of Privacy-Preserving Inference of Transformers (PiT) increases, a hybrid protocol that integrates Garbled Circuits (GC) and Homomorphic Encryption (HE) is emerging for its implementation. While this protocol is preferred for its ability to maintain accuracy, it has a severe drawback of excessive latency. To address this, existing protocols primarily focused on reducing HE latency, thus making GC the new latency bottleneck. Furthermore, previous studies only focused on individual computing layers, such as protocol or hardware accelerator, lacking a comprehensive solution at the system level. This paper presents APINT, a full-stack framework designed to reduce PiT's overall latency by addressing the latency problem of GC through both software and hardware solutions. APINT features a novel protocol that reallocates possible GC workloads to alternative methods (i.e., HE or standard matrix operation), substantially decreasing the GC workload. It also suggests GC-friendly circuit generation that reduces the number of AND gates at the most, which is the expensive operator in GC. Furthermore, APINT proposes an innovative netlist scheduling that combines coarse-grained operation mapping and fine-grained scheduling for maximal data reuse and minimal dependency. Finally, APINT's hardware accelerator, combined with its compiler speculation, effectively resolves the memory stall issue. Putting it all together, APINT achieves a remarkable end-to-end reduction in latency, outperforming the existing protocol on CPU platform by 12.2x online and 2.2x offline. Meanwhile, the APINT accelerator not only reduces its latency by 3.3x but also saves energy consumption by 4.6x while operating PiT compared to the state-of-the-art GC accelerator.