HOLLYWOOD — After the release of multidisc collections on Columbia, Capitol and Reprise, Frank Sinatra fans might think they own all the Sinatra recordings that can possibly exist.

They’d be wrong.

On June 4, Warner’s Reprise and Turner Classic Movies Music labels will jointly release “Sinatra in Hollywood (1940-1964),” a six-CD, 160-track anthology consisting of the artist’s output for movies over the decades.

Amazingly, most of the material has never been released before. Just as amazing, given the array of Hollywood studios involved, was the effort to snare the rights for the compilation.

Producer Charles Granata, author of “Sessions with Sinatra,” explains: “Although many of Sinatra’s greatest hits came from his film songs — songs like “The Lady Is a Tramp,” “All the Way,” “My Kind of Town” — the versions that we’re used to hearing are not the original film performances. They were re-recordings.”

Singer-songwriter Michael Feinstein calls the collection “the first of its kind, because it collates songs from the complete span of a single entertainer’s musical film career. It’s a document of our entertainment history as well as a time.”

Granata and Grammy-nominated producer Didier C. Deutsch, with the backing of the Sinatra estate, spent the past two years making the rounds of movie studios, university collections, recorded-sound archives and private sources to unearth every original Sinatra recording made for the movies.

Deutsch estimates costs for the project at around $100,000 ,not including clearance or licensing fees, which could easily match or top that figure.

The compilation was no easy task. During his Hollywood career, Sinatra worked for MGM, RKO, Warner Bros., Fox, Paramount, Columbia, Universal, Goldwyn and United Artists.

But the consolidation of media companies worked in the project’s favor. With Warner/Reprise/Turner backing the project, the MGM, RKO and Warner songs — which constituted a large portion of the material needed — were readily accessible.

Still, Sinatra’s prickly reputation across town, thanks to numerous studio run-ins, initially meant closed doors. Deutsch recalls one studio exec telling him he discarded a query letter because “the name Frank Sinatra means problems.”

Sinatra’s granddaughter, music supervisor A.J. Lambert, served as liaison between the family and the studios. Issues of rights and ownership were complicated by studio mergers, lack of paperwork or simple disinterest at the studio-business level.

“A lot of times they often don’t know exactly what rights they have,” Lambert says. “It was time-consuming.”

Paramount wanted its legal research paid for; Universal was missing key audio elements. Eventually, though, all but Fox agreed to clear tracks for the collection.

Granata says his biggest disappointment is the inability to include Sinatra’s unreleased version of “If I Loved You” from “Carousel,” the 1955 Fox musical that Sinatra quit before filming began.

Fox officials declined comment, but sources say the studio couldn’t reach agreement with the Sinatra estate over the scope of rights to be granted for the project.

So they didn’t get Sinatra’s original 1954 “Three Coins in the Fountain” from Fox either. The set does include his commercial recording of “Three Coins,” and a newly discovered (though incomplete) eight-minute version of “Soliloquy” from “Carousel,” recorded at Capitol in 1955.

But fans of The Voice will have a lot to savor.

“Practically everything from the early years” will surprise listeners, Deutsch says, including the very first track, a Frank Loesser number called “Dolores” that was cut from Sinatra’s first movie role in Par’s 1941 “Las Vegas Nights.”

And, it turns out that Sinatra recorded nine numbers from “Finian’s Rainbow” for a never-finished 1954 animated version of the Burton Lane-E.Y. Harburg musical, including duets with Louis Armstrong and Ella Fitzgerald.

One Holy Grail for Sinatra collectors eluded Granata and Deutsch: “Farewell, Amanda,” the Cole Porter tune that Sinatra sings in the 1949 Tracy-Hepburn classic “Adam’s Rib.” Both finally concluded that, with the long-ago destruction of the original nitrate optical-film recording at MGM, it probably doesn’t exist.