This paper focuses on constructions of MDS self-dual codes from (extended) generalized Reed-Solomon (GRS) codes. Let <inline-formula> <tex-math notation="LaTeX">$q = r^{2}$ </tex-math></inline-formula> be an odd prime power. We show that, there exists a <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>-ary self-dual (extended) GRS code for each even length in the range <inline-formula> <tex-math notation="LaTeX">$[{2r,3r-3}]$ </tex-math></inline-formula>, and for each singly even length in the range <inline-formula> <tex-math notation="LaTeX">$[3r-1,4r]$ </tex-math></inline-formula>. This extends the only known consecutive range <inline-formula> <tex-math notation="LaTeX">$[2,2r]$ </tex-math></inline-formula> to <inline-formula> <tex-math notation="LaTeX">$[{2,3r-3}]$ </tex-math></inline-formula> for this case. Furthermore, our general constructions provide many MDS self-dual codes with new parameters which, to the best of our knowledge, were not reported before.