Large language models (LLMs) have shown promise on summarization tasks, but they often produce hallucinations, which are unsupported or incorrect statements that limit their reliability in specialized healthcare applications. We introduce Hallucination Detection Guided Self-Refinement (HDSR), an inference-time method that leverages hallucination detectors to guide iterative summary revisions toward factual corrections. Building on this, we propose HDSR for Preference Learning (HDSR-PL), which converts detector-guided refinement trajectories into preference pairs for model finetuning. Extensive experiments show that our methods substantially reduce hallucinations for Llama and Gemma models in summarizing real-world clinical notes from MIMIC-IV-Note v2.2. For example, HDSR reduces 24% and HDSR-PL reduces 48% hallucinations in Llama-3.1-8B-Instruct. Importantly, both methods preserve summary fluency, coherence, and relevance according to human expert and LLM-Jury evaluations. Together, these results demonstrate that detection-informed refinement and preference learning offer an automated solution for improving factual faithfulness in clinical summarization.