If the foreign worker is subject to the annual numerical limit, or H-1B cap, then the earliest start date is October 1st of that fiscal year.
If the foreign worker is outside the United States then s/he must obtain an H-1B visa before s/he will be admitted to the United States in H-1B status. If the foreign worker entered the United States in a different non-immigrant visa category (i.e. visitor); was present in the United States in valid non-immigrant status on the first date of validity of the H-1B petition; and a request for a change of status from the non-immigrant status to H-1B status was requested, then the foreign worker could begin employment on the first date of validity of the H-1B status (October 1st).
If the foreign worker is not subject to the H-1B cap (was previously granted H-1B status in the last 6 years) and is present in the United States in valid H-1B status then the foreign worker can begin employment as soon as the H-1B petition is filed with the USCIS (this assumes that the foreign worker’s new employment is in the same or a similar occupation).