This paper proposes a new key scheduling method for AES-256 that enhances security against related-key differential cryptanalysis. Unlike existing approaches, our goal is to minimize code modifications (i.e., avoiding additional operations) to maintain compatibility with existing implementations. This is crucial because extensive changes to widely deployed implementations may introduce new vulnerabilities and implementation errors. By thoroughly analyzing the processing steps in software implementations utilizing AES-NI, we introduce two approaches to improve security and efficiency. The first approach modifies only the arguments of a specific function, reducing the required security rounds from 13 to 9 without affecting performance. The second approach removes two lines of code, which not only reduces the required security rounds from 13 to 8 but also improves software speed. Both methods enhance security while maintaining high compatibility with existing AES-256 implementations.