I have the same situation.

回答: Google: H-1B portabilityform1402008-10-03 14:31:17

As far as I know, I think the answer is you are legal
to work and don't need to go abroad.

But there is a few restrictions:
1. You must get the receipt before you can work for
the new employer.
2. You must file H1B transfer before your current H1B
expires.

There may be a few other things you can google out.

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