Thanks to Jose for suggesting this question.
It depends. You can only transfer your H-1B if you have been in status at all times in the past. The situation is easy if you immediately find a new job. Otherwise, you would have to change to another status, such as B-2, to remain in status. If you neglected this and continued to rely on your H-1B, then most likely you will indeed have to leave the USA (because you violated your status). You may be able to use the old H-1B visa to return, or you may need to get a new one, depending on circumstances.
The ultimate way to find out if you have to leave the USA is to wait for the USCIS decision. If there is an I-94 attached, they forgave you being out of status, and you can stay in the USA. This used to be routine a few years ago, but USCIS now uses a zero tolerance policy and rarely issues new I-94s in such a situation.
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