Can a wife be taken back after a minor irrevocable divorce (baynunah sughra) based on the old marriage contract and her consent, without formal divorce proceedings in court?

If you divorce your wife less than three times, and her waiting period (iddah) expires before you take her back, she becomes irrevocably separated from you (baa'in). You do not have the right to take her back except with a new contract. The meaning of a new contract is that her guardian (wali) marries her to you in the presence of two witnesses and with a new dowry (sadaq). There is no condition for the contract to be written or to be authenticated in courts.