Seat on elephant's back

One word substitute is Howdah.

Saddle: a seat fastened on the back of a horse or other animal for riding, typically made of leather and raised at the front and rear.
Howdah: (in South Asia) a seat for riding on the back of an elephant or camel, typically with a canopy and accommodating two or more people.
Lounge: lie, sit, or stand in a relaxed or lazy way.
Hoof: the horny part of the foot of an ungulate animal, especially a horse.