Presenting users with diverse responses from foundation models is crucial for enhancing user experience and accommodating varying preferences. However, generating multiple high-quality and diverse responses without sacrificing accuracy remains a challenge, especially when using greedy sampling. In this work, we propose a novel framework, Synthesize-Partition-Adapt (SPA), that leverages the abundant synthetic data available in many domains to elicit diverse responses from foundation models. By leveraging signal provided by data attribution methods such as influence functions, SPA partitions data into subsets, each targeting unique aspects of the data, and trains multiple model adaptations optimized for these subsets. Experimental results demonstrate the effectiveness of our approach in diversifying foundation model responses while maintaining high quality, showcased through the HumanEval and MBPP tasks in the code generation domain and several tasks in the natural language understanding domain, highlighting its potential to enrich user experience across various applications.