I ride up their a couple of times each summer. I can suggest some trails for you ride, But I've always brought my own horses and have no idea of who to hire if you didn't have your own horses.
I suspect that any of the guides services should be able to set you up to ride any trail that you are willing to pay for the hours on. You are just asking for something more than their average 2 hour tourist ride.
There are a lot of trails with in easy trailering distance of Jackson, I see outfitters unloading horses and heading off from the various trail heads. So you don't have to do ride where they leave from a corral at the visitors center.
This is a trail into Bechler Meadows in the South West Corner of Yellowstone
This is a trail around Two Ocean Lake that leaves from the Jackson Lake Lodge. The lodge has a corral of horses and takes out dude string rides. But we just park in the parking lot and ride the same trail.
You get some tremendous views of the Grand Tetons
We usually camp about an hour south of Jackson either along Granite Creek, Cabin Creek or near Bondurant, We can ride the local national forest or trailer on up into the Grand Teton or Yellowstone Parks
This is Granite Creek just south of Jackson.
Here we have ridden up Granite Creek close to where the Cache Creek trail from Jackson hole hooks up with it.