In Jonathan Coulton's "Sticking It To Myself," the first word or phrase in one line (bolded) often also serves as the last word in the next line without repetition:
And I heard everything you said
Those things to try to get inside my head
Is full ...
Sticking it to my-
That's not the only thing I lack
Just my own gun against my
Back down now and let this hostage
This is as opposed to the chorus of The Wanted's "Glad You Came," which does something similar but repeats the word on the next line:
Turn the lights out now
Now I'll take you by the hand
Hand you another drink
Drink it if you can
Can you spend a little time?
Time is slipping away
Away from us, so stay
Stay with me, I can make
Make you glad you came
What The Wanted's "Glad You Came" does by repeating the last words of one line in the start of the next line is called anadiplosis, but the name of the rhetorical scheme or literary device I'm seeking is that of the first example where the words at the end of one line are not repeated but instead themselves become the words that start the next line. Is there a name for this rhetorical scheme or literary device? If so, what is it?