Self-dual MDS codes over finite fields are linear codes with important cryptographic and combinatorial applications. In this correspondence, a different approach is proposed to obtain Hermitian self-dual MDS codes from (extended) generalized Reed-Solomon (abbreviation to GRS and EGRS) codes. The necessary and sufficient conditions of a Hermitian self-dual (extended) GRS code are presented. With the help of this new method, we revise and improve the results in Niu and Yue <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> (IEEE Communications Letters, 23(5): 781-784, 2019). Furthermore, we suppose that there exist Hermitian self-dual (extend) GRS codes over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {F}_{q^{2}}$ </tex-math></inline-formula> if and only if their code length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n\leq q+1$ </tex-math></inline-formula> .