I agree with my colleague. Company A can file a new H-1 transfer petition for you.
This answer is of a general nature and should not be relied upon as final, nor is it intended as legal advice. Consult with a qualified attorney before making any legal decisions. Gen Kimura, (832) 247-6932.
Company A would have to repetition for you as a Change of Employer H1B petition.
An attorney-client relationship is not formed by my responses to questions on Avvo. My responses are not intended to be legal advice and must not be relied upon as legal advice.