The white discharge may be food. If there's a yellow/white material inside the socket, its probably just part of the natural healing process. It sounds like he's doing better. I would keep an eye on the situation but I don't think any more antibiotics will be necessary. The swelling in his cheek may be a cellulitis so it will take some time for it to clear out. Since the offending tooth is gone, the source of infection is gone and the body should be able to clear it out.
Could the discharge be food debris or pus?
I would probably get it checked out. The expert will be better able to advise.