How does movie music staff know all the less known songs?

Sometimes they just pick the right person to do the choosing.
Some people just really know their subject.

^{This doesn't fully answer the question, but bear with me ;)}

A friend of mine is a record librarian at the BBC.
He can tell you not only the release date & record label of any song you ask him, he can also describe the cover if it has one [not all singles do, of course] & how many different variants the actual printed label was issued in.

He stores his own vast record collection in the simplest way for him to go directly to any one - chronologically.

He isn't that knowledgeable because he's a record librarian - he's a record librarian because he's that knowledgeable.

He's also a musician, & that almost total recall of all those records means he can also play just about any of them, if asked.

If you want to present interesting but obscure incidental or featured music to your audience, you ask the right person for what they think would fit your brief.

Almost everyone knows of some obscure song by a group that never especially hit the mainstream.

It's not so much a matter of them knowing all songs and choosing from them, as much as, between the group, they know a few obscure ones, some of which may fit with the theme and mood of the film.

Maybe they know only Top 40 pop songs, except for an obscure half dozen, between their music team. Having a half dozen obscure songs that make it onto a soundtrack might give an impression of greater knowledge of "hidden" gems that does not reflect reality. You can't really know how much of their knowledge base is represented by that.

They do research, it is quite obvious someone will have in their mind of certain songs, won't they? Some songs as old as the First World War are still known today.

Songs are generally not lost to the world as people may think, but they are just phased out and are not listened to anymore...this does not mean they don't exist anymore. There is a number of multi-billion people in the world...and a lot of old people still know of songs from back then.

Movie staff will just look up certain songs because they in fact can and they have access to a wide variety of music, old and new, of all genres etc. It isn't really rocket science to how they find seemingly lost songs..they ask people they know.

They use Primary sources:

They use search engines, other movies, music history, record disks (they still exist), old day devices, tapes and very old people.

They also use secondary Sources:

They use things like websites, articles online (not from the original author) and plenty of other things you could possibly think of.

They could also advertise online for certain music and maybe a reward for the given song or genre of music.

It is really quite easy to obtain old and almost long forgotten music, they just think of a way and get it even if it is supplied to them by someone.