Actually, Nadine - it is easier to get a permit hiking down from the North Rim.
90% of tourists visit the South Rim, only 10% the North Rim - probably similar for inner canyon hikers.
There are 3 possible campgrounds for transcanyon hikers: (South Rim to North Rim)
Indian Gardens halfway down from the SR, Bright Angel at Phantom Ranch, and Cottonwood halfway to the NR.
There are also pricey cabins at Phantom Ranch and a less-expensive bunkhouse.
Indian Gardens and Bright Angel are booked solid for months in advance.
Cottonwood often has openings a few weeks out and cancellations the morning of.
By getting two nights at Cottonwood, you can hike down in late afternoon -
Spend a full day exploring the inner canyon with a light pack -
Then hike out during the cool of the third morning.
Done it many times.
Good luck and enjoy.
Pic - Hiking down the North Kaibab Trail along the sheer redwall