Alison Pill plays the role of Zelda Sayre Fitzgerald -- yes, Montgomery's own -- in Midnight. Despite being Canadian, she has a really good Alabama accent. This makes me wonder, which actress not born in the south has the most authentic southern accent? I don't mean Texas or Hillbilly, I mean Southern as in Louisiana to the Carolinas (or "old Florida").
The best, to my recollection, is Jessica Lange. She started her movie career with an awful performance in 1976's King Kong but recovered with good reviews in the 1980s. Specifically, I remember her as Maggie in a 1985 production of Cat on a Hot Tin Roof for television. Lange drawled out "I feel all the time like a ca-at on a hot tin ru-uf", providing the obligatory two syllables for "cat" and "roof". Startingly impressive! See for yourself on YouTube (at the 6:43 mark). Lange was born in Minnesota, by the way.
As for native actresses doing southern accents, there are none better than Holly Hunter and Reese Witherspoon.