SDCL § 46A-10A-113: Appointment to fill vacancy.

If any vacancy occurs in the membership of the board of trustees between the annual elections, the remaining members of the board may fill the vacancy by appointing a person with the required qualifications. The appointed person shall qualify in the same manner and hold office until the next annual election when a successor is elected. If all places on the board become vacant, then a new board shall be elected in the same manner as was the initial board.

Source: SL 1997, ch 256, § 16.