Assalam Alaykum,When exactly is the range of time regarded as the last/Third period of the night most prescribed and recommended for prayer answering?

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

The method of determining the last-third part of the night is to know the beginning of the night and its end, and then divide it into three parts. The third part that is before Fajr is the last-third part of the night. The beginning of the night begins at sunset and ends at dawn (Fajr). This varies according to different times of the year, sometimes the night is long and sometimes it is short.

Shaykh Ibn Baaz was asked in Majmoo’ Al-Fataawa:

“It is mentioned in a Hadeeth, "Allah descends to the first heaven (sky) every night during the last-third of the night.” When does the last third of the night begin and when does it end?

He answered:

“There are many Mutawaatir narrations (Mutawaatir are Ahadeeth that are narrated by a group of narrators from a group of narrators, generation after generation and so forth, and they are all trustworthy people and it is absolutely impossible for them to agree on a lie) from the Prophet that confirm the descent of Allah, among which is the saying of the Prophet, sallallaahu 'alayhi wa salaam:"

“Our Lord descends to the first heaven every night at the last-third of the night and says: ‘Is there anyone who is invoking Me so that I may respond to his invocation? Is there anyone who is asking of Me, that I may grant him his request? Is there anyone asking for My forgiveness, so I may forgive him,' [Muslim]

The beginning and end of the third (of the night) differ according to the season. If the night is nine hours long then the time of descent begins at the onset of the seventh hour, until dawn comes. If the night is twelve hours long then the last third begins at the beginning of the ninth hour (from sunset) until dawn comes, and so on, depending on how long or short the night is in each place. And Allah is the One who grants guidance.” [End of quote]

Allah knows best.