The Grey Cock is a bit untypical in some respects. As a rule, there's no supernatural element to the song at all; the lover has to leave at cock-crow, not because he's dead, but because he is quite the opposite, and has to go to work:

The cocks are crowing, love, I must be going,
For we are servants and we must obey.

He tends to be cold and wet and not terribly pleased, but he's alive alright, and has had a rather nice time even if it didn't last as long as he'd hoped:

The wind it did blow and the cocks they did crow
As I tripped over the plain, so very plain, so very plain,
So I wished myself back in my true love's arms
And she in her bed again.

The discovery in 1951 of Cecilia Costello and her set of The Grey Cock, ghost and all, led a lot of people to think the song to be in all its forms a revenant ballad, from which in most cases the supernatural element had disappeared; however, as Hugh Shields pointed out in his study, The Grey Cock: Dawn Song or Revenant Ballad? (Ballad Studies, ed. Emily B. Lyle, 1976), the supernatural verses in this case are all borrowed from a completely different song, the 19th century Irish broadside Willy-O. Indeed, in the fairly rare cases when Grey Cock variants do have ghostly presences, they can generally be traced to another, separate source; often Sweet William's Ghost (of which Willy-O, mentioned earlier, seems to be a broadside re-write).

A rather extreme example is Willy's Fatal Visit (Scottish, of course!) where a normal night-visit ends with the hero's dismemberment -on his way home- by the vengeful shade of his ex-girlfriend. Here, we actually have bits of several quite distinct songs bolted together (perhaps by Peter Buchan, who first published it): tradition has cut it back to the essentials again, and Jeannie Robertson sang only the concluding part.