Hilary got here at midday and Nicholas arrived soon after.
after = adv.
♥♦♣♠ NOT A TEACHER ♥♦♣♠
That's it - normally, after (adverb) is preceded by another adverb like soon, shortly, straight, etc.
Mind you, the adverb after left alone in a sentence is common, I reckon, in informal speech, with the meaning of later.