Widespread frustration with rigid touch-tone Interactive Voice Response (IVR) systems for customer service underscores the need for more direct and intuitive language interaction. While speech technologies are necessary, the key challenge lies in routing intents from user phrasings to IVR menu paths, a task where Large Language Models (LLMs) show strong potential. Progress, however, is limited by data scarcity, as real IVR structures and interactions are often proprietary. We present a novel LLM-based methodology to address this gap. Using three distinct models, we synthesized a realistic 23-node IVR structure, generated 920 user intents (230 base and 690 augmented), and performed the routing task. We evaluate two prompt designs: descriptive hierarchical menus and flattened path representations, across both base and augmented datasets. Results show that flattened paths consistently yield higher accuracy, reaching 89.13% on the base dataset compared to 81.30% with the descriptive format, while augmentation introduces linguistic noise that slightly reduces performance. Confusion matrix analysis further suggests that low-performing routes may reflect not only model limitations but also redundancies in menu design. Overall, our findings demonstrate proof-of-concept that LLMs can enable IVR routing through a smoother, more seamless user experience -- moving customer service one step ahead of touch-tone menus.