So I’m going to recommend forgoing best of jazz samplers as they can be a bit of a mixed bag, That includes the Ken Burns Jazz Series. Not knowing your preferences for early jazz, big band, vocals etc. I had to make some assumptions. Also, some of the earliest recordings make be lacking in “fidelity”. However, they are worth pursuing as the create a basis for the jazz you may enjoy going forward. I’ve left out quite a bit. As I type this more come to mind. Pursuing this list will result in the beginnings of a top notch jazz library that you may continue to build on, versus a sampler that may sit on the shelves. By the way thank you for limiting your request to pre-1960, other wise my recommended list would be a bit intimidating in size and scope.
(Charles Mingus Jazz at Massey Hall is a must have.)
So here it is and keep in mind these are albums I would start with:
Traditional
Louis Armstrong – The Complete Hot Five and Seven Recordings
Big Band
Duke Ellington – The Okeh Ellington 1927 - 1931
Duke Ellington and his Famous Orchestra - Fargo, North Dakota November 7, 1940
Duke Ellington - The Blanton Webster Band
Count Basie and his Orchestra - April in Paris
Count Basie – The Complete Decca Recordings
Count Basie – The Complete Atomic Basie
Bennie Goodman – The Famous 1938 Carnegie Hall Jazz Concert
Vocals
Billy Holiday – Lady Day: The Best of Billie Holiday
Ella Fitzgerald – Ella Fitzgerald Sings the Rodgers and Hart Songbook
Ella Fitzgerald – Ella Fitzgerald Sings the Irving Berlin Song Book
Ella Fitzgerald – Ella Fitzgerald Sings the Cole Porter Song Book
Combos
Lester Young – The Complete Aladdin Recordings of Lester Young
Charlie Parker - The Complete Dial Recordings
Charlie Parker The Complete Savoy Recordings
Coleman Hawkins – The Indispensable Coleman Hawkins “Body and Soul” 1927 -1956
Sarah Vaughan – Sarah Vaughan with Clifford Brown
Clifford Brown – Clifford Brown and Max oach
Sun Ra - Greatest Hits Easy Listening for Intergalactic Travel
Sonny Rollins – Saxophone Colossus
Sonny Rollins – Night at the Village Vanguard
Thelonious Monk – Genius of Modern Music Vol. 1
Thelonious Monk Genius of Modern Music Vol. 2
Thelonious Monk – Brilliant Corners
Thelonious Monk - Thelonious Monk with John Coltrane
John Coltrane – Giant Steps
John Coltrane – Blue Train
John Coltrane – My Favorite Things (1961)
Miles Davis - Round About Midnight
Miles Davis - Kinda Blue
Cannonball Adderley – Somethin’ Else
Charles Mingus – Ah Um
Charles Mingus – Jazz at Massey Hall (This is a Super Group of BeBop with Charlie Parker, Dizzy Gillespie, Bud Powell, Charles Mingus, and Max Roach)
Bud Powell The Amazing Bud Powell Vol 1.
Bud Powell – The Amazing Bud Powell Music Vol. 2
Wes Montgomery – The Incredible Jazz Guitar of Wes Montgomery
Stan Getz – Storyville Vol. 1
Art Pepper – Art Pepper Meets the Rhythm Section
Art Blakey – Moanin’
Dave Brubeck Quartet – Time Out
Ornette Coleman – The Shape of Jazz to Come
Ornette Coleman – Change of the Century