Earlier quoted context omitted.
H1B Transfers don't count against the H1B quota. If you still have time left on your H1B (i.e. approved I797 petiton) and have not violated the 'leave immediately if out of job' rule, it is possible to have a new employer file a petition for transfer of the I797 (H1B) In fact, in such cases, you don't even need a new visa stamp for the transferred H1B. As long old H1B stamp has not expired, you can present this H1B a…
This doesn't appear to be an H1B transfer. Even if it were some sort of H1B transfer, I believe by leaving the US he might have abandoned his H1B. So I think the OP better get his immigration status verified by his new employer, because the last thing he wants to do is get a 10 year ban at the border.
Basically, you must get a brand spanking new H1B to start a new job. However, if you have already recently gotten a H1B, then you can skip the quota, and get it approved pretty much immediately. This is commonly referred to as a "transfer" even though it is not technically a transfer.