Is it permissible to pray between the first and second calls to Fajr prayer, and when does the last third of the night end?

The texts indicate the legitimacy of two adhans for Fajr: the first before Fajr for preparation, and the second at its arrival, signaling the entry of the prayer time. What is permissible during the night is permissible between the two adhans. The Prophet (peace and blessings be upon him) said: "Let not the call of Bilal—or he said, his adhan—prevent any of you from taking your suhoor, for he calls—or he said, he gives the adhan—so that your standing worshipper may return [to rest]." And the last third of the night ends with the advent of Fajr, not with the first adhan, based on the saying of the Prophet (peace and blessings be upon him): "Our Lord, the Blessed and Exalted, descends every night when the last third of the night remains, and says: 'Who is there to ask Me that I may give him? Who is there to supplicate to Me that I may answer him? Who is there to seek My forgiveness that I may forgive him?' until Fajr appears."