How can we move a subpanel without replacing the supply cable?
This is pretty routine, and I've done a lot of it.   

Slap a 120mm (4-11/16 if you really want to say all that) square deep junction box where the subpanel used to be.  Plumb 3/4" preferably 1" EMT or rigid conduit from the new subpanel to the aforementioned box.  (Do all this in reverse; coordinate the location of subpanel and junction box so the conduit run is easy, then run the conduit.) 

Bring the #6 cable into the 120mm box with a cable clamp.  Attach the cable ground wire to the junction box shell.  The metal conduit is the ground path, and you don't need to run a ground wire unless you really want to.  

Get some #6 stranded THWN-2 wire. Run three wires: 


neutral (must be white or gray wire) 
L1 (any color but green) 
L2 (any color but green, bonus style points if different color than L1)


If L1 or L2 are white or gray, you must mark them near both ends with a color of paint or tape other than white, gray or green.  

These wires run from the junction box to the subpanel lug.   Inside the junction box they are wire-nutted or spliced  to the corresponding wire in the cable.   

If you want to splice other cables into that box too, use 1" conduit and don't go crazy - no more than four circuits per conduit, and watch your junction box fill cubic inches, on a deep 120 you're nearly out already.   They do make box extensions.