Does cohabitation during the waiting period after a divorced wife is taken back verbally necessitate a new marriage contract?

If the wife is taken back (reconciled with) before the expiry of her 'iddah (waiting period) with an explicit statement, then she returns to the husband's marital authority without the need to renew the contract. However, if the reconciliation occurs after the 'iddah has ended, it is not valid, and she becomes a stranger to him, necessitating a renewal of the contract with all its pillars. Reconciliation is valid from an absent husband, just as divorce from him is valid.