I might suggest the trail from the Yellowstone Picnic area that goes to the top of the ridge on the opposite side of river from Tower Falls. If you have four people, I would try the Pelican Valley Trail if it is open. This area doesn't open until 9:00am and you must be out by sundown, and travel in a group of four or more people. I have done seven miles round trip and it gets you out into a nice valley similar to the Hayden, but very few people. Good chance to see bears or wolves.
Another neat one is to hike behind Grand Prismatic Spring via the Fairy Fall trail. When you get behind it, use one of the social trails and climb the hill. This gives you a nice view of the Spring. Bring your widest angle lens, a tripod, a neutral density filter, and a polarizer and have a good time. This is about 2 miles round trip.
At Mammoth you can do the Beaver Ponds trail. This has a bit of an elevation change and as I recall is a loop of about 5 miles. Probably won't see any beaver, but maybe a black bear or if you are lucky a grizzly. Be careful of the elk in the area at this time of year, the bulls are pretty dangerous and cranky.
The one bad thing about this time of year is the weather. I have already had reports of snow in the area, so pack appropriately. I will be in the park most of the day on the 19th, and returning on the 23rd. Also, I cannot stress this enough, buy and carry bear spray with you. It can be found at most of the general stores in the area, if you are flying. If not stop at a Cabelas, Bass Pro, Sportsmans Warehouse, etc and get yourself some.
Have fun and if you have specific questions feel free to PM me. Tom R. and Rik are also great resources for Yellowstone and the Tetons.