Now in addition to Indiana Jones and the Calendar of the Sun, which had you working with a non-playable Indiana Jones in a race to find the calendar (which was said to be able to control eclipses) against fellow archeologist Doctor Forrestal (the skewered skeleton in Raiders of the Lost Ark, shown here alive and well several years earlier), there is Warriors of the Sky.
In Warriors of the Sky, Indiana Jones is asked by his friend Harold Oxley (from Indiana Jones and the Kingdom of the Crystal Skull) for help in finding the Temple of the Sky. Indiana Jones gets a lead on a piece of the temple in Peru, so you begin your search and get assistance from Harold Oxley over two-way radio.
The authentic Indiana Jones feel of the Indiana Jones missions is thanks to Hal Barwood (author of Indiana Jones and the Fate of Atlantis and Indiana Jones and the Infernal Machine), who is credited in this game in the area of "narrative design".
It's still a Zynga game (so there's still energy depletion, requirements of requests to friends to receive items, and the optional ability to purchase in-game items with real money), but in my opinion it's worth the social game hassle to experience the story.