In the 3rd Year, Prisoner of Azkaban, Harry leaves the Dursley's home and before getting picked up he sees the Grimm which turns out to be his Godfather Sirius Black. So Sirius Black has already escaped from Azkaban in the beginning of the book. Later on, when Sirius Black and Harry are being attacked by the Dementors by the lake, the alternate Harry releases a Patronus Charm that saves both Harry and Sirius. They then fly Buckbeak and break out Sirius from Azkaban. Taking into consideration that Hermione had to return by the end of the final bell ring and only went back in time 3 turns, my question is: How is it possible to break out Sirius Black in the middle of the story when Sirius Black had broken out prior to the beginning of the 3rd year?
You're mis-understanding the situation.
Sirius escapes from the Azkaban in the summer.
At the end of the story Sirius is locked in Hogwarts, awaiting a Dementor to come and "kiss" him. So Harry and Hermione are not flying Buckbeak to Azkaban just to the tower at Hogwarts that Sirius is locked in.
‘Sirius is locked in Professor Flitwick’s office on the seventh floor. Thirteenth window from the right of the West Tower. - Prisoner of Azkaban