L M montegomery I"m pretty sure wrote another book called the Story Girl that I still enjoy rereading today, It would be good for her age. And i agree Anne of Green Gables would be ok, Part of me wants to say wait on that since the other books in the series get more mature and i'm the kind of kid that likes to finish a series all at once not year by year.
I was going to suggest the Lion the Witch and the Wardrobe too, but again like The Secret Garden It might be better to wait just a bit.
There are Marc Brown's --Arthur series chapter books that would be much closer to her developmental age.
If she likes Mysteries, Ron Roy has Calendar Mysteries characters are first graders, A to Z mysteries characters are second graders or third, my kids love them, He has a series set in DC the Capital Mysteries too.
If she is more into fantasy, Any of the Rainbow fairy books would be good for her. and there are Tons.
If she is a pet lover Ellen Miles has a series called the Puppy Place --those books are sooo wonderful, I think she has a series called Kitten Korner too. but i have a hard time finding them.
As for the classics, i think Little House in the Big woods is the one to start with, and save the others until she has a bit more life experience and maturity under her belt.
My son is a whizz kid reader and his teacher read him Tales of a fourth grade nothing ( beware the revealing of Santa) and he devoured all of those books, his favorite was Super Fudge. again, i sort of his his teacher would have waiting on the tales of a fourthgrade nothing, but...
Happy reading!!!