on the next day: the day after the one the account has just described
This is a plain calendar marker, and it belongs to storytelling. Seventeen times in the New Testament it opens a scene with the force of "and the day after that," moving the account forward by exactly one day: the chief priests going to Pilate the morning after the crucifixion, Jesus walking out of Bethany hungry, Paul leaving Lystra the morning after he had been stoned and left for dead.
Every time it appears, the word "the" stands in front of it, because the word for "day" is simply left unspoken. The phrase says, in effect, "on the [day] after." That is why English Bibles render it with a phrase, "the next day" or "on the morrow," rather than with a single word.
It is not the word for "tomorrow." αὔριον looks forward from wherever the speaker is standing ("do not worry about tomorrow"), while this word looks back from a point the story has already reached. Luke never uses it in his Gospel but reaches for it constantly in Acts, where he alternates it with two other ways of saying the same thing, ἑξῆς and ἐπιοῦσα, apparently just for variety.
John makes the most deliberate use of it. Three times in his opening chapter the word ticks off one day after another (John 1:29, John 1:35, John 1:43), chaining together a run of days that many readers count as a week leading up to the wedding at Cana. Later it marks the morning after the feeding of the crowd, and the morning of the entry into Jerusalem. In each case the word itself claims nothing more than "the day after"; the weight comes from how carefully the writer has strung those days together.