i. Usage is always substantive and with the dative article, τῇ ἐπαύριον, meaning on the next day:
The next day, when they had come out from Bethany, he was hungry. [Mark 11:12 WEB]
Καὶ τῇ ἐπαύριον ἐξελθόντων αὐτῶν ἀπὸ Βηθανίας, ἐπείνασεν.
ii. τῇ ἐπαύριον is exactly equivalent to ἐπὶ τὴν αὔριον:
On the next day, when he departed.... [Luke 10:35 WEB]
Καὶ ἐπὶ τὴν αὔριον { ἐξελθών ♦ - }....
It happened in the morning.... [Acts 4:5 WEB]
Ἐγένετο δὲ ἐπὶ τὴν αὔριον....