If Hermione was born in September, it means her 1st year at Hogwarts was when she were 12, so when did she got her Hogwarts letter? still at 11?, or just at a random day?
According to J.K. Rowling, Hermione was born September 19, 1979, meaning she was almost 12 when she started at Hogwarts. I would assume she got her letter that summer, meaning she was 11 at the time, and was still 11 when she started at Hogwarts, turning 12 about 18 days into her time at Hogwarts.