I am facing the same issue. I google it, finding some indian students have
successfully worked it out. The new H1B is COS, so they ask school for a new
I20 with your new employee, which is the work authorization in the rest of
cap-gap.
I inquire the official of international office. They are not 100% sure, but
they think they can do it if I offer a new H1B petition receipt or approval
notice.
My understanding, the cap-gap is an extension of OPT, but requires school's
permission (I-20 with employee name).