From what I read, the J-1 requires you to immediately leave the country upon completion of the contract. Of course, one can still apply for another VISA but it does not give you any buffer zone where you can continue to reside and work in the US in the meantime.

For this reason I heard some case where people opt to do a secondary MSc rather than do a post-doc, to get an F-1 VISA instead. Apparently the F-1 gives you three years time to get an H1-B.