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 is His slave and Messenger.
First of all, we say that even if the best time for Qiyaam (voluntary night prayer) is the last third of the night, its time begins after the ‘Ishaa' prayer. Therefore, whoever performs Qiyaam at the beginning of the night (i.e. right after the ‘Ishaa' prayer) gets the reward of Qiyaam.
‘Aa’ishah said: “The Prophet, sallallaahu ‘alayhi wa sallam, observed Witr prayer in every part of the night, at the beginning, middle and in the last part. So, his performance of Witr extended until dawn.” [Al-Bukhari and Muslim]
What you have mentioned about Abu Bakr beginning his Qiyam right after the ‘Ishaa' prayer, we have not come across any authentic text about it. However, it was reported that he used to perform the Witr prayer after the ‘Ishaa' prayer for fear of missing out on performing it, and then he would sleep and then wake up and stand up in prayer until the Fajr prayer.
Abu Daawood reported that the Prophet, sallallaahu ‘alayhi wa sallam, asked Abu Bakr: “When do you pray Witr?” He replied: “I perform Witr in the first part of the night.” Then, he, sallallaahu ‘alayhi wa sallam, asked ‘Umar: “When do you pray Witr?” He replied: “I perform Witr in the last part of the night.” Thereupon, the Prophet, sallallaahu ‘alayhi wa sallam, said to Abu Bakr: “This one has followed it with caution,” and he said to ‘Umar: “This one has followed it with strength.” [End of quote]
Besides, Abdur-Razzaaq reported in Al-Musannaf that Abu Bakr and ‘Umar spoke about Witr in the presence of the Prophet, sallallaahu ‘alayhi wa sallam. Abu Bakr said: “As to me, I sleep after having prayed the Witr prayer. Then if I wake up, I pray an even number of Rak‘ahs until the Fajr prayer." ‘Umar said: “I only sleep after having prayed an even number of Rak‘ahs, and then I pray the Witr prayer just before dawn (i.e. just before the time of the Fajr prayer starts)." Thereupon, the Prophet, sallallaahu ‘alayhi wa sallam, said to Abu Bakr: “You are cautious,” and he said to ‘Umar: “You are strong.”
As regards your statement that the last third of the night is the best time for Qiyaam, then this is the view of a group of jurists. However, the view of the majority of the scholars is that the best time of the night is the third of the night after the middle of the night, and not the last third of the entire night. The Fiqh Encyclopedia reads: “The scholars differed in opinion in regard to the best time for Qiyaam: The majority of the scholars are of the view that the best time is the fourth and fifth sixth parts of the night as Abdullaah ibn ‘Umar narrated that the Prophet, sallallaahu ‘alayhi wa sallam, said: “The most beloved prayer to Allaah is the prayer of Daawood and the most beloved fasting to Allaah is that of Daawood; he used to sleep half the night, stand in prayer for a third of it and sleep a sixth of it; and he would fast a day and not fast the next." [Al-Bukhari].” [End of quote]
Therefore, if we presume that the hours of the night from sunset to dawn are six hours, then the best is for a person to sleep for three hours, and then stand for Qiyam for two hours, and then sleep again for one hour; and this is in conformity with the Qiyaam of Prophet Daawood .
Allaah Knows best.