Indeed, it's a pretty canny move by PTA to blend songs that are (more or less) from the era with Greenwood's work, as it adds another textural layer that keeps audiences off balance during what is already a delightfully off-kilter picture. The songs gently offer insight to the plot and characters at various points throughout the film, so here's a bit of a warning: there are mild spoilers in the list of songs below, particularly when it comes to one song near the end that you might be best going in cold to experience.
Otherwise, if you have seen the movie, here's everything that caught your ear. Samples below including Greenwood's astounding "Baton Sparks."
"The Master" Complete Tracklist
"Baton Sparks" from "48 Responses To Plymorphia" - Jonny Greenwood
"Get Thee Behind Me Satan" - Ella Fitzgerald
"Dahil Sa Iyo"- written by Miguel Velarde Jr.
"Sweet Sue, Just You" - Noro Morales
"Overtones" - Jonny Greenwood
"You Go To My Head" - Larry Clinton & His Orchestra
"Don't Sit Under The Apple Tree (With Anyone Else But Me)" - Madisen Beaty
"Dancers In Love" - Duke Ellington
"Lotus Blossom" - Duke Ellington
"I'll Go No More A-Roving" - Philip Seymour Hoffman
"Children Of The Family" - Eban Schletter
"Celebration Solitaire" - Eban Schletter
"A-Tisket, A-Tasket" - Melora Walters
"No Other Love" - Jo Stafford
"The Deep Boo Sea" - written by Winston Sharples
"On A Slow Boat To China" - Philip Seymour Hoffman
"Changing Partners" - Helen Forrest