While moving to jazz could be offshore drilling in a bottomless well of covers, I'll restrict myself to jazz covers of rock songs, beginning with the astounding Brad Mehldau.
One of Mehldau's many versions of Paranoid Android (thought I should add to the small number of Radiohead covers). He has a few recordings of the track that are over twenty minutes, but I'll go with one under ten.
Mp3 Brad Mehldau / Paranoid Android (Radiohead Cover)
Here's a video of Brad with his trio doing Exit Music (For a Film):
And a solo cover of Elliott Smith...
Mp3 Brad Mehldau / Bottle Up and Explode (Elliot Smith Cover)