Ruling on taking back a wife divorced with a revocable divorce after she has exited the waiting period

This is a letter that reached us from Muhammad Nasr al-Din Mustafa, a Sudanese resident in the northern part of Yemen, in the city of Sana'a. The brother Muhammad has several questions in his letter. He says at the beginning: He has a sister whose husband divorced her three years ago with a single divorce. They have one son who is about nine years old, and he is the only one Allah kept alive for them among their sons and daughters whom Allah the Mighty and Majestic took to Himself. He says: I am trying to restore the waters to their channels and reconcile between them. Is reconciliation permissible in this case? And is it permissible for the woman to return to the protection of her husband? How is the return to be done? Please inform us, may Allah reward you.
Answer: If the woman mentioned was divorced with only a single divorce and he did not divorce her more than that, he may take her back as long as she is in the waiting period. If she has exited the waiting period, it is permissible for him to return to her with a new marriage contract with its conditions considered valid by the Shari'ah. That is, he proposes to her, and if she agrees, her guardian marries her to him with the mahr (bridal gift) on which they mutually agree, and this is done in the presence of two witnesses.
As for if she is in the waiting period, meaning three menstrual cycles have not yet passed for her, or she is still in the waiting period, he takes her back without a contract. He says: Bear witness, so-and-so and so-and-so—two witnesses—that I have taken back my wife so-and-so, and that is sufficient. There is no need for a contract as long as she is in the waiting period. If she menstruates and three menstrual cycles have not come to her, she is in the waiting period, so he takes her back without a contract. Likewise, if she is of advanced age and her menstruation has stopped, her waiting period is three months. If the three months have not passed for her, he takes her back without a contract. However, after she exits the waiting period, a new contract is necessary. This is like the rest of the men: he proposes to her guardian and to her. When she agrees and her guardian agrees, and they mutually agree on a mahr that satisfies her, then the contract is completed. The guardian says to him: I have married you to my daughter so-and-so, if her father is present, or my sister so-and-so, if it is her brother. He says: I have accepted this marriage, in the presence of two just witnesses, and the matter is done. Yes.
The presenter: May Allah reward you.