No. If you already hold an H-1B status in the United States, you have already been counted towards the cap (unless you were previously employed by an institution of higher education or an entity affiliated with an institution of higher education). Because you have already been counted, you do not need to be counted again. Your current H-1B is not “transferred” to your new employer; rather, the employer needs to petition for a new H-1B. Your new employer may file an H-1B petition on your behalf at any time, provided the assignment you will be performing is a professional assignment and all the additional H-1B requirements are met.