Employees in H-1B status must notify the Center for Global Health prior to leaving employment, regardless of the reason for the end of the employment (termination, voluntary resignation, sponsorship from another H-1B employer, etc.). MUSC is required by law to report this information. H-1B employees are also required to provide CGH with forwarding contact information.
There is no official “grace period” for H-1B status after the period of authorized stay in the US ends. You should plan to depart the US immediately at the end of your employment (authorized period of stay) if you have not received a change of status or if another employer has not filed a new H-1B petition on your behalf.