The answer is, there is no answer. Not one that we will be able to figure out anyway.
In each of the first 5 books the DADA teacher was a previously unintroduced character. And even in book 6, the new teacher (who ended up teaching potions, as we all know) had not been introduced in any of the prior books. However, if Jo follows her initial pattern the DADA teacher/new teacher will be introduced before Harry meets him or her at Hogwarts.
- Harry first met Professor Quirrel at the Leaky Cauldron in Philosopher's Stone.
- Professor Lockhart was introduced at the book signing at Flourish and Blots in Chamber of Secrets.
- Lupin rode on the Hogwarts Express in Harry's train car in Prisoner of Azkaban.
- Mad-Eye Moody was introduced when Arthur Weasley had to go to his house in Goblet of Fire.
- Umbridge was present at Harry's trial in Phoenix.
- Professor Slughorn was introduced the night Harry and Dumbledore went to Slughorn to convince him to teach at Hogwarts.
So, small consolation though it is, if my theories on the publication date are correct we should know who the new teacher at Hogwarts is (DADA or otherwise) sometime early in the wee smalls of July 7, 2007.