Attic and Koine

[Gryllus Maior]

Plenty of interaction between the two in later antiquity (which includes the "Biblical Koine"):

The changes of Greek during the Roman period along the aforementioned two major trends of diachronic development and conservatism resulted in the three varieties of Greek employed in Late Antiquity prose: (a) low register Koine, used in private texts (e.g. epistolography) and some literature (e.g. early Christian writings); (b) high-register Koine, used in official inscriptions and literature (e.g. by Plutarch); (c) Atticism (employed by rhetors, e.g. Aelius Aristides and Lucian). However, there is a mutual influence between the higher registers, and thus a compromise between 'high-register' Koine and Atticism is to be observed."

--Late Antiquity Prose in the ENCYCLOPEDIA OF ANCIENT GREEK LANGUAGE AND LINGUISTICS, p. 317.

Particularly, the book of Acts, 2 Peter and Hebrews are written in higher register Koine, and one finds Atticisms in them...

 

[The Real John Milton]

Lol

 

[Gryllus Maior]

Classical → Attic, used in literary master works of the 5th and 4th c. BCE (philosophy, tragedy,oratory), enjoyed a long-standing prestige over the other Greek dialects and a leading role in the development of the Hellenistic Koine and ultimately of Modern Greek.

--Linguistic Variation in Classical Attic in the ENCYCLOPEDIA OF ANCIENT GREEK LANGUAGE AND LINGUISTICS, p. 364

 

[Lucian]

Plenty of interaction between the two in later antiquity (which includes the "Biblical Koine"):

The changes of Greek during the Roman period along the aforementioned two major trends of diachronic development and conservatism resulted in the three varieties of Greek employed in Late Antiquity prose: (a) low register Koine, used in private texts (e.g. epistolography) and some literature (e.g. early Christian writings); (b) high-register Koine, used in official inscriptions and literature (e.g. by Plutarch); (c) Atticism (employed by rhetors, e.g. Aelius Aristides and Lucian). However, there is a mutual influence between the higher registers, and thus a compromise between 'high-register' Koine and Atticism is to be observed."

--Late Antiquity Prose in the ENCYCLOPEDIA OF ANCIENT GREEK LANGUAGE AND LINGUISTICS, p. 317.

Particularly, the book of Acts, 2 Peter and Hebrews are written in higher register Koine, and one finds Atticisms in them...

There's some dispute as to whether there is such a thing as 'Koine', or at least whether it's a useful category. But you're right to highlight the element of continuity between earlier and later Greek.