You can apply for H-1B extension, by requesting for "Nunc-Pro-Tunc" relief. USCIS may forgive if the lapse was not too much.
If they do not wish to forgive, they would at least approve the petition and you can use that to get a visa stamp abroad.
I believe you can also show the AP and show the approved H-1B at the POE to indicate that once admitted, you will be working for the H-1B employer. In your case, this does not seem to be a safer route (perhaps) and if yes, it is better to get a new visa stamp.
