Agentic workflows, where multiple AI agents collaborate to accomplish complex tasks like reasoning or planning, play a substantial role in many cutting-edge commercial applications. These workflows depend critically on the prompts used to provide the roles models play in such workflows. Poorly designed prompts that fail even slightly to guide individual agents can lead to sub-optimal performance that may snowball within a system of agents, limiting their reliability and scalability. To address this important problem of inference-time prompt optimization, we introduce ProRefine, an innovative inference-time optimization method that uses an agentic loop of LLMs to generate and apply textual feedback. ProRefine dynamically refines prompts for multi-step reasoning tasks without additional training or ground truth labels. Evaluated on five benchmark mathematical reasoning datasets, ProRefine significantly surpasses zero-shot Chain-of-Thought baselines by 3 to 37 percentage points. This approach not only boosts accuracy but also allows smaller models to approach the performance of their larger counterparts. This highlights its potential for building cost-effective and powerful hybrid AI systems, thereby democratizing access to high-performing AI.