Rhapsody released an upgraded version of its iPhone app that allows users to cache songs on the device so that they can be played even when the device is out of wireless coverage.

That gives the company bragging rights for being the first to offer an on-demand music streaming app with that feature in the U.S. Spotify offers it with its smartphone apps overseas, but it has yet to go live in this country. MOG also plans to introduce a mobile app with offline caching, but it is not expected to be available until next month.

But the company did have to qualify the upgrade. As of this week, only playlists built by Rhapsody subscribers will play offline. The ability to cache individual albums and songs won’t be added until June.

The ability to play music even when the device is not connected to a WiFi or cellular network is a crucial element of on-demand music streaming services. While coverage has certainly improved in recent years, the reality is no device is 100% connected at all times. The first time users can’t play music as a result of this lack of connectivity, the value of on-demand streaming drops while that of pay-to-own files increases.

The new Rhapsody app is available only for the iPhone, with plans to add support for Android devices later this summer, and for the iPad when the multitasking feature of the iPhone 4.0 OS is released.

Rhapsody says its existing app has been downloaded 1.5 million times.

Update: Thumbplay Music was actually the first to come out with an on-demand streaming service with caching capabilities earlier this year for the BlackBerry. The Rhapsody app is the first to do so for the iPhone.