In the following night prayer, should one recite from where the reading ended in the prayer, or from where the reading ended outside the prayer, with the intention of completing the Quran?

It is permissible for one who intends to complete the recitation of the Qur'an to read it both during prayer and outside of it, and from a mushaf (copy of the Qur'an) or from memory. This reading counts towards the completion. He can start from where he left off in his recitation. There is no harm in reading from the mushaf during the night vigil prayers (Qiyam al-Layl). Ibn Qudamah reported from Ahmad that it is permissible in supererogatory prayers (nawafil) but disliked in obligatory prayers (fara'id) except out of necessity, whereas Ibn Hamid made no distinction between supererogatory and obligatory prayers in terms of permissibility. The evidence for this is what was narrated from 'A'ishah, may Allah be pleased with her, that her slave used to lead her in prayer while reading from the mushaf, and the statement of Az-Zuhri that the best of the tabi'in (successors) used to read from the mushafs.