The Second Temple was destroyed in 70 CE and with it the sacrificial system, including the rites of the Day of Atonement that the priestly texts prescribe. What is striking in the rabbinic response is the absence of crisis about atonement itself.
The sources attach to Rabban Yochanan ben Zakkai a teaching given as the Temple burned, that acts of loving kindness now atone in the place of sacrifice, and the argument is made from a line in Hosea in which God says that he desires mercy and not sacrifice. The move is not presented as an emergency measure. It is presented as a reading of what the prophets had already said the offerings were for.
The liturgy of the Days of Awe states the replacement directly, in the formula that repentance, prayer and charity avert the severity of the decree. Maimonides set out the requirements of a complete repentance in the Laws of Repentance, and that treatment, with its insistence that the test of repentance is behaving differently when the same opportunity returns, has remained the standard statement.
One later custom needs separating from all of this because the shared root invites confusion. Kapparot, performed by some before Yom Kippur, involves swinging a fowl over the head with a formula transferring one's deserts to it, after which the bird or its value goes to the poor. It has been criticised by rabbinic authorities since the medieval period, some of whom regarded it as resembling a foreign practice, and it is defended by others as an established custom. Whichever view is taken, kapparot is a custom and kapparah is the concept, and the atlas keeps them apart.
Read against the Christian theories, the post-Temple settlement makes one thing plain. A tradition that had a sacrificial system for atonement, and lost it, did not conclude that atonement had become impossible. That is worth stating, because arguments which move from the Israelite offerings straight to the necessity of the cross pass over a tradition that read the same texts and did not draw that conclusion.