All perfect praise be to Allaah, the Lord of the worlds. I testify that there is none worthy of worship except Allaah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
If a person offered the ‘Asr prayer and then remembered that he had not prayed Thuhr yet, he has to pray Thuhr and is not required to pray ‘Asr again. Reoffering the ‘Asr prayer in this case is recommended, considering the view that holds that it is obligatory to abide by the prescribed order of prayers. This view is adopted by the Hanafi, Maaliki, and Hanbali scholars, who also said that this order should be observed when one has to perform missed prayers and the current one.
As evidence for their view, they cite the statement of the Prophet, sallallaahu ‘alayhi wa sallam: "Whoever forgets a prayer or sleeps until its prescribed time has elapsed should pray it whenever he remembers it." Other wordings of the narration read: "Whoever forgets a prayer, then its due time is whenever he remembers it." This narration implies that the time of performing a missed prayer is when the person remembers it. Hence, performing the current prayer before making up the missed prayer means that the current prayer is prayed before its due time and this is impermissible.
So the scholars' deduction of the Hadeeth is contrary to what you aim to deduce from it, based on the fact that the due time of a missed prayer is the time when it is remembered. Taking into account that observing the prescribed order of prayers is obligatory, the prayers that follow the missed one in order should be redone if they were prayed before it. There is evidence from the Sunnah supporting this deduction, such as the Hadeeth narrated on the authority of Ibn ‘Umar who said that the Prophet, sallallaahu ‘alayhi wa sallam, said: "Whoever forgot a prayer and did not remember it until he was praying with the Imaam, let him complete his prayer and, when he finishes it, let him offer the prayer that he forgot. Then let him redo the prayer which he offered with the Imaam."
Moreover, it is narrated that once, during the Year of Al-Ahzab, the Prophet, sallallaahu ‘alayhi wa sallam, was praying Maghrib and upon finishing the prayer, he asked: "Has anyone of you seen me pray ‘Asr?" They said: "No, you have not prayed it O Messenger of Allaah.” Thereupon, he ordered the Muezzin to call the Iqaamah (second, small call to prayer) and he prayed ‘Asr and then reoffered the Maghrib prayer." [Ahmad] So this is the correct course of action in this case, since the Prophet, sallallaahu ‘alayhi wa sallam, said: "Pray as you have seen me pray."
Allaah Knows best.