In the rapidly evolving environment of wireless medical sensor networks (WMSN) and the internet of medical things (IoMT), remote medical support has seen unprecedented advancements. It is essential that the data relayed from the sensors must be trustworthy and unaltered, and that the sensors themselves are genuine. Wireless networks, however, have inherent vulnerabilities. In addition, since WMSN is directly linked to patients' lives, its continuous availability is crucial. Considerable efforts have been made to maintain the integrity and authenticity of such data. However, many studies have failed to address the problem of a single point of failure (SPOF). This issue has been particularly detrimental to patients who require ongoing management. To address this issue and ensure the protection of the authenticity and integrity of patient data, we suggest the implementation of an authentication scheme based on blockchain technology. In 2022, Yu et al. introduced a blockchain-integrated authentication and key generation scheme for WMSN using Physical Unclonable Functions (PUFs), effectively addressing the SPOF problem by conducting mutual authentication through smart contracts without relying on centralized servers. Our research found that this scheme inadvertently shared critical parameters, including challenge-response pairs and important private keys, with the blockchain network, making it vulnerable to various breaches. We present an enhanced protocol designed to mitigate these security challenges. By limiting the data interaction with smart contracts and ensuring only relevant parties access crucial parameters, our approach reduces the risk of public information disclosure on the blockchain. This not only mitigates the SPOF issue but also efficiently helps in prevention of physical attacks. We prove that our proposed system prevents known security vulnerabilities through informal and formal analysis using the Scyther, Proverif, and BAN logic. Furthermore, the proposed scheme offers 67.37% reduction in computation costs and 3.67% in communication costs, presenting an efficient and secure solution for WMSN in the IoMT landscape.