Aluminium foil won't help, because it's static magnetic fields which are the problem and aluminium foil doesn't shield magnetic fields. You would need something with a high permeability, like Mu-metal, if you wanted to shield it.
I was thinking of hacking the clamp mechanism and remove the isolation between the two clamps, so that the core material can actually touch. That should reduce the leakage and the drift quite a bit, I reckon.
Taping off the current signal and move it outside with a BNC connector is another thing on my list, but haven't gotten around to play with the thing yet ... so much to do, so little time.