Another thought is to fly into Billings and return from Jackson thought this becomes more expensive. Or fly into Billings and leave from Bozeman which is slightly less expensive. Thanks for your thoughts.

I vote for sure a night at Old Faithful Inn and then Roosevelt Lodge/cabins would get my second vote. You might consider Lake Hotel too and finally Canyon area. I am sure the lodging is fine at Mammoth area, but it is my least favorite part of the park. I would suggest you spend a night in Grand Tetons too. Colter Village, Jackson Lake Lodge, Signal Mountain are some of the choices there. If you can spend more than a couple of days in Yellowstone/Tetons. Amazing place.

Last trip we flew into Billings, drove over Beartooth Pass into the park, stayed at Old Faithful Inn four nights as base, drove out to Cody for one night for the rodeo (not in September though) and museum, then back to Billings. Doing it again I'd probably split the lodging in the park into two night stays possibly with Roosevelt lodge to save some driving.

I think I was posting the same time as Spirobulldog. Completely agree Mammoth area was underwhelming, expected a whole lot more.

Thanks.
Too bad Roosevelt is closed by Sept 2. Everybody seems like Roosevelt.
We could stay at Old Faithful and Canyon maybe a few days each.
Does the drive from Cody back to Billings take you back over Beartooth Highway ? What would Cody be like in mid -September ?

My Dad lives in Cody and we fly in 4 or more times a year so we've seen Yellowstone/GTNP multiple times over the last 8 years.
Our favorite itineary is to fly into Cody, drive the loops thru Yellowstone (staying 2 nights) and then drive south thru
Yellowstone, into GTNP, stay 1 night in GTNP at Jackson Lake then drive to Jackson Hole stay 1 night in Snake River Lodge. You can then fly out JAckson or lopp back thru Yellowstone to Cody overnight and then fly out Cody next day.
Be aware Yellowstoneand GTNP book room reservations thru 2 different companies

Airports:

Bozeman MT--small airport Delta,small rental car co, good for North Yellowstone entrance. Good for overnight in Big Sky ski country: won't be any snow tho lodging will be cheaper. Check a map for access to Beartooth HWY

Billings MT--bigger regional airport with more airline/flight options. Bigger rental car choices. Solid 2 hour very boring (watch the speed limits drop into tiny nondescript farm towns-speed traps & tickets big) drive to Cody West Entrance to Yellowstone. Ck map for alternate Billings to Beartooth HWY route. Provison w/ice chest, ice and breakfast & lunch suplies at Costco or Wlmart Superstore.

Cody, WY--small airport Delta and United airlines only right at East Entrance Yellowstone. Few rental cars Hertz, Avis, 2 other very cheap co. Fun frontier town with fantastic Wild West Buffalo Bill Museum, Big Horn Sheep exhibit, Irma Hotel, saloon (featured on Amazing Race).
Nice overnight (Americinn Inn,Bubbas BBQ w/salad bar in parking lot) nice Chineese Buffet, Walmart to provision w/ice chest, ice, Subway or breakfast/lunch supplies.)

Jackson, WY--small airport, small rental car, South access to JAckson Hole ski area, GTNP.
Smaller the airport, the bigger the price.

Be aware food inside both parks are concession and not that great for the price.

Be aware rental car co are all private own franchises and they can charge what they want, won't honor discounts except AAA and even if you book an SUV, they may not have it in stock. They will often not allow drop off no matter how much you pay: and will charge you for ANY damage they find so inspect car before you take it off lot and take a picture of any damage. You also may want to have extended AAA roadside servcie as if you have a flat tire You have to fix it and miles in between service is far. Be sure to keep your gas tank topped off too. Not alot of servcies in Parks and are far in between and expensive.

On all roads watch for animals: Antelope or Pronghorn, Mule deer most commen and will kill you or your rental car so watch out esp if they are outside the fence. Have seen BEAR, moose, buffalo, eagles, hawks, porcupine, coyotes, cows, raccoons etc.

Yellowstone PArk
East entrance from Billings or Cody my preference. This entrance does close for the Winter so check closure dates. Be aware if it snows, it may be difficult due to incline & need for chains, as they will close the road for safety.
You enter East entrance with steep incline, but fantastic views of river below and come upon the giant lake inside Yellowstone. From the Lake you can go North or West or South to GTNP. Would agree at least 2 night stay in 2 places Yellowstone, one near geysers Old Faithful Inn, one your choice.
http://www.yellowstonenationalparklodges.com/

IMHO, GTNP is by far the best park. We go here every time now and just drive thru Yellowstone to Cody to visit my Dad. Less people, more and variety critters, less traffic jams more wide open expanse of the Grand Tetons and the Snake River. More off the main road driving to remote places, more diversity of hiking and things to do. I'd stay at Jackson Lake Lodge. Fantastic views of Tetons from giant lounge windows. Best for dinner but not gourmet.
http://www.gtlc.com/

For additional night I'd stay at JAckson Hole Snake River Lodge w/indoor pool, spa and gourmet dining room.
http://www.snakeriverlodge.com/default-en.html

Gosh I did not even consider Cody a possible gateway so thats another good choice.
Because of time constraint we would only do Yellowstone.
I agree the Tetons are also special.
So it is still a toss up between BIL BZN and now COD. The cheapest fares are to SLC for us. Can't decide whether the savings in $$ is worth the 5-6 hour OW drive to Yellowstone, therefore still looking to see which nearby gateway city is the best for us.

Thanks to everybody for advice. I think we have been convinced we should add two nights and include the Tetons. Flying into Bozeman and flying out of Jackson Hole seems to be a reasonable way to accomplish this . We plan to drive to Red Lodge and spend our first night there; drive the Beartooth Hwy the next day and spend three nights in Canyon then another three nights at Old Faithful. The last two nights will be in Jackson Hole.
Surprised some lodgings are already booked for September. Looking forward to a great time !