"Dumb Supper" (1964) or "Call Me From the Valley" (original) mentioned only 2 or 3 songs, depending on the version of the story:
  - Dumb Supper - pp. 116, 121
  - In the Pines (original version only)
  - A Farewell to Arms - p. 126

Despite being materially the same story, this chapter has been published under two different names: Dumb Supper (1964 Ballantine Books), or Call Me From the Valley (original). The chapter mentions fairly few songs in comparison to most of the other chapters. The original version of the chapter includes John singing a fragment of "In the Pines" exactly as they appeared in Shiver In the Pines while walking in the late night rain, while the 1964 version omits this song.

Dumb Supper

"Oh, call me sweetheart, call me dear,
Call me what you will,
Call me from the valley low,
Call me from the hill."

"I hear you as the turtle dove
That flies from bough to bough,
And as she softly calls her mate,
You call me softly now...."

I believe that this song is based on the folk ballad Mary Ann/Turtle Dove/Fare Thee Well. You can read more about the song here, here. You can read a version collected in Carmen, NC here, and a very good breakdown of the songs related to it from Mudcat (though several other threads on the song exist as well).