Missed recognition of named entities while de-identifying clinical narratives poses a critical challenge in protecting patient-sensitive health information. Mitigating name recognition errors is essential to minimize risk of patient re-identification. In this paper, we emphasize the need for stratified sampling and enhanced contextual considerations concerning Name Tokens using a fine-tuned Longformer BERT model for clinical text de-identifcation. We introduce a Hidden in Plain Sight (HIPS) Markov-based replacement technique for names to mask name recognition misses, revealing a significant reduction in name leakage rates. Our experimental results underscore the impact on addressing name recognition challenges in BERT-based de-identification systems for heightened privacy protection in electronic health records.