No. Your current SSN is fine. I still use the SS Card I got when I came here on a J-1 in 1993. No problems at all. I haven't been to a SS Office since 1993.Being on H-1B, I already have a SSN. But in one of the posts I found someone commenting that, there is some work to be done at the SSN office after getting the EAD before I can start a job without any sponsorship. Is that right?