Accessing medical literature is difficult for laypeople as the content is written for specialists and contains medical jargon. Automated text simplification methods offer a potential means to address this issue. In this work, we propose a summarize-then-simplify two-stage strategy, which we call NapSS, identifying the relevant content to simplify while ensuring that the original narrative flow is preserved. In this approach, we first generate reference summaries via sentence matching between the original and the simplified abstracts. These summaries are then used to train an extractive summarizer, learning the most relevant content to be simplified. Then, to ensure the narrative consistency of the simplified text, we synthesize auxiliary narrative prompts combining key phrases derived from the syntactical analyses of the original text. Our model achieves results significantly better than the seq2seq baseline on an English medical corpus, yielding 3%~4% absolute improvements in terms of lexical similarity, and providing a further 1.1% improvement of SARI score when combined with the baseline. We also highlight shortcomings of existing evaluation methods, and introduce new metrics that take into account both lexical and high-level semantic similarity. A human evaluation conducted on a random sample of the test set further establishes the effectiveness of the proposed approach.