I got the SSN when I came here on H1B
So, there was a restriction on the SS Card which says something like \'Can work only with INS Authorization\'. But, since I got my passport stamped, I got those wordings removed by reapplying at the SSA office. It doesn\'t matter, if the restriction is still there, but as I said in my earlier posting, you dont have to provide ur GC when u change employer, if the wordings / restriction is removed from the SS card.
Hope this helps.