It depends what you're looking for in terms of an authentic experience. I've hiked over 2,000 miles in the canyon spanning 33 years. I was lucky to have a father who was a canyon hiking freak who thought it fun to take his 5 year old son down there. Good thing CPS wasn't there.
The South Kaibab, North Kaibab and Bright Angel Trails are considered the "corridor" trails and as such...are by no means remote or "backcountry" in feel. They're not easy hikes, but in places, those trails are 9 feet wide, have rest shacks, railings, etc. They also have 1000 tourists at the top all doing their best to clog the trail. Then, there's the mule poop and pools of pee. Corridor hiking has its merits, but you have to take it for what it is. Same for Phantom RANCH, not CREEK. Phantom Ranch is really neat, has great food, and offers a great deal of luxury for its location, but it's not "remote" and you do have to share it with quite a few others. Phantom Creek on the other hand is a fine hike, but not for a solo.
There are also tons of other amazing trails that are far more remote, but no more or less difficult. I've hiked all over the Alps, Rockies, Sierras, Atlas Mountains and the wilds of Alaska. The Canyon might be one of the most amazing wilderness areas I've ever had the pleasure to visit. If you go to the more remote areas (just a few miles off the corridors) you will only ever see a handful of other backpackers as opposed to maybe hundreds on the corridors. Regarding the true backcountry trails, some only see a dozen hikers a season, but due to the permitting control, even a solo hiker is easily located if trouble strikes and is overdue. I've been on trails that only see a couple hikers
per decade which to me is amazing considering the proximity to Vegas, Phoenix, etc. What an amazing place.
One of my favorites is a hike down the New Hance trail to Hance Rapids. Then over to Cottonwood Creek below Horseshoe Mesa. The hike out is up the Grandview Trail. You get a river camp. You get a Tonto Plateau hike and an exit along one of the most spectacular ridgeline hikes in the park. I usually do that in 3 days, but 4 would be a tad more relaxed. When I say favorite, I mean to say I've done that loop at least 6 times and still adore it. It's a perfect slice of all that the park has to offer. You walk past ancient indian granaries, massive rapids, huge dunes and volcanic slabs. You even hike through one of the most amazing mining operations in canyon history. Bar none the best 3-5 day jaunt in the park.
Regarding season, September is still bloody hot. October and November are best. November and April are my favorite canyon months. I used to do extremely remote water surveys for the Park Service and my one really ugly trip due to lack of water and high heat was in September. Blah. Not fun. I also used to guide multi-day trips for the Grand Canyon Institute and September would often have some of our clients suffering more than they had anticipated.
Regarding Rim to Rim hikes, those too can be fun, but I think there are far, far more exciting hikes down there. Rim to Rim gets more objective oriented and less about the splendor of the canyon.
Shoot me a PM if you have questions. I'm a recovering Grand Canyon Junkie. I spent so much time down there in the late 90s, the Park put me to work! Wonderful place.