Asaph sets the stage with a crisis of faith in Psalm 73 and the book ends with the darkest Psalm in all the Bible, Psalm 88. In the midst of the dark book of the Psalms, book 3, there is what Spurgeon called the sweetest Psalm and the center of the center of the center of that ( Ps 81:8 is 'if only My people would listen to Me and also arguably listen to me about the day of atonement alluded to in the middle of the two trumpets blown in vs ps 81:3 )
We look at the flow as Asaph, Sons of Korah, David, Heman and Ethan navigate through the darkness finding hope and resolution in God. 'Hope in God' one generation tells another in Psalm 77, even in the midst of these struggles.
Psalm 78 is a historical psalm and recounts failures of Israel particularly regarding temptations of food, power and worship. Jesus will go through these same temptations in the dessert yet prevail.
Psalm 81 is arguably the center psalm of the books. Center book is book 3 and in book 3 8 are before 8 after making psalm 81 the center psalm with the center verse being verse 8 'if only My people would listen to Me' and one can press it further with the two trumpets in verse 3, the center of those is the day of atonement. Book 3 is about crisis of faith and the other psalms and books revolve around that to resolve the crisis. There is hope since psalm 81 speaks of the man at God's right hand. Jesus listens to God where his people did not and is the one to resolve the crisis.
Books 2 and 3 are written largely by Levites and poetically follow Exodus and Leviticus. It is fitting that in the final two Psalms portend the ultimate sacrifice, the death of the Messiah, the anointed, the seed of David in psalm 88 with an upturn/resurrection in Psalm 89.
An lesser issue of book 3 concerns who Asaph is since the temples destruction would be future to the Asaph of David's day. Asaph was a seer so perhaps it's prophesy or perhaps from the musician group of levites founded by Asaph continuing to write psalms?