You may not use a Section 8 voucher to receive rental assistance in two different units at the same time.
If you are the head of household, you may port your voucher to the area you will be moving to, as long as you have lived in your unit for one year, or were a resident when applying to the Section 8 waiting list. The housing authority that covers the area you are moving to must be currently absorbing vouchers, so contact that office for more information. If the housing authority is not absorbing, you may have to port to a different housing authority that covers a nearby area. Contact the housing authority that manages your voucher for more information. However, if you do port, your family would no longer receive rental assistance in the unit you are currently living in.
If you are not the head of household, you may move out of the unit, but you cannot use the head of households voucher to cover your new unit. You may apply for housing choice voucher assistance through the housing authority that covers the area you are moving to.