As you correctly mention, the levitation effect is due to the pinning of magnetic flux lines in a type II superconductor.
The magnetic flux $\Phi$ that goes through the SC is a function of the external field. This flux is made up of vortices that each carry a flux quantum $\Phi_0$. These vortices are evenly distributed through the SC. If the magnetic field at the SC increases, more flux lines will need to move into the SC. This motion is hindered by pinning of the flux lines.(1)
Now in order to levitate, you will need an inhomogeneous magnetic field: If the SC followed gravity, it would move closer to the coil and therefore to a place of higher magnetic field. That would mean vortex motion, which is however hindered by pinning. The SC will rather stay at a place of constant magnetic field, which explains the levitation.(2)
In your experiment, the argument goes the other way around: if you crank up the current, the SC again wants to stay at a place of constant magnetic field because otherwise, it would need to overcome the pinning forces. So it will rise up to find a place with the same field that it saw before you increased the current.
(1) There's a nice movie that shows how more flux lines enter the SC as the magnetic field is increased: http://www.youtube.com/watch?v=vwFm7d_0GsA
(2) Consider for example a superconducting train as in http://www.youtube.com/watch?v=TeS_U9qFg7Y: Along the tracks, the magnetic field is homogenous, so that the train can freely move forward. Perpendicular to the tracks, however, the field is inhomogenous, so that the train does not sink down or come off the track.