Augmented Large Language Models (LLMs) enhance the capabilities of standalone LLMs by integrating external data sources through API calls. In interactive LLM applications, efficient scheduling is crucial for maintaining low request completion times, directly impacting user engagement. However, these augmentations introduce scheduling challenges due to the need to manage limited memory for cached information (KV caches). As a result, traditional size-based scheduling algorithms, such as Shortest Job First (SJF), become less effective at minimizing completion times. Existing work focuses only on handling requests during API calls by preserving, discarding, or swapping memory without considering how to schedule requests with API calls. In this paper, we propose LAMPS, a novel LLM inference framework for augmented LLMs. LAMPS minimizes request completion time through a unified scheduling approach that considers the total length of requests and their handling strategies during API calls. Recognizing that LLM inference is memory-bound, our approach ranks requests based on their consumption of memory over time, which depends on both the output sizes and how a request is managed during its API calls. To implement our scheduling, LAMPS predicts the strategy that minimizes memory waste of a request during its API calls, aligning with but improving upon existing approaches. We also propose starvation prevention techniques and optimizations to mitigate the overhead of our scheduling. We implement LAMPS on top of vLLM and evaluate its performance against baseline LLM inference systems, demonstrating improvements in end-to-end latency by 27%-85% and reductions in TTFT by 4%-96% compared to the existing augmented-LLM system, with even greater gains over vLLM.