a small connecting word that signals what comes next is a new step, development, or shift in what the writer is saying
δέ never stands first in its clause; it always comes second, leaning on the word ahead of it. It carries no meaning of its own. What it does instead is give the reader a cue: take what comes next as a new step or development from what just came before. That is why English translators reach for "but," "and," "now," or "then" depending on the situation. None of those English words is what δέ means; each is a stopgap that captures one effect a new step can have when it follows a particular kind of statement.
Because δέ is a cue rather than a word with content, the old labels "contrasting" and "continuing" are not meanings of the word at all. They simply describe how two clauses happen to relate in a given verse. When the two clauses are pulling against each other, the new step sounds like "but." When it is just the next thing that happens, it sounds like "and" or "now." The contrast or the addition lives in the clauses themselves, not in δέ. One major grammarian put it plainly almost a century ago: there is no built-in idea of opposition or contrast in the word.
δέ is the fourth most common word in the New Testament, turning up roughly 2,937 times. How often a writer uses it is a matter of style: Luke and Acts, Matthew, Hebrews, and Paul use it freely, while Mark uses it rarely and leans on καί instead. That heavy use of καί is a sign of Hebrew influence on how the book was composed. Outside the Bible, everyday Greek letters and documents from the same era use δέ in exactly the same way, which shows it is ordinary Greek of the period, not a special biblical habit.