I am not sure about new H1B. But here is what happened to me. I decided not to wait for the offer from Company A, and took the one from Company B. I did H1B transfer from old job to company B. A few works after I sent out the H1B transfer, offer from Company A arrived. My attorney filed another H1B transfer for me, not sure from which to which.

After I wrote down the above, I realized my experience might not be helpful at all.

Check with your attorney or your office of international office. I bug them once in a while. :)