Wednesday, Thursday and maybe Friday might be good. The Weekend could be tough, but you could get lucky.
Be flexible. Bright Angel Campground might be booked, but consider going a little farther to Cottonwood. Or carrying extra water from Phantom Ranch and going up and out on Clear Creek to dry camp. The Rangers at the Back Country office are usually pretty helpful.
Be really flexible. If you have a little extra money, check in at the desk in the Bright Angel Lodge and see if any raft trips have cancelled reservations that day for the Phantom Ranch Dorms.
If someone ahead of you in line has just snagged a Corridor permit for less than 6 people, maybe you can charm your way and get them to add you to their permit.
If you have water purification skills and tools, consider the Threshold trails. Hermit or Grandview.
It is going to be warm, so take care and hydrate.