If your H-1 was approved w/ COS, then you need to start working on H-1 from COS effective date (typically Oct 1).
Your full-time employer can file cap-exempt petition for you (aka H-1 transfer) and you can work for them on H-1 thereafter.
Another option is to take an action to invalidate your COS and continue working on L-2 EAD. You can do this by either filing COS from H-1 to L-2 (after Oct 1) or leaving US and returning on L-2 visa after Oct 1.
Lastly, if COS was not approved, then you can continue working on L-2 EAD for the new full-time employer.