There are “Big Apple caps” (also called “newsboy caps”) or “Big Apple hats” or “flat caps.” The name “Big Apple” comes from the 1937 Big Apple dance craze.
     
The “Big Apple cap” is a flat cap, and is also called the “newsboy cap” because it was worn by New York City boys (called “newsies”) who sold newspapers. The cap has also been worn by workmen and sportsmen since at least the 1800s, and has been called by many other names. The cap became popular again around 1970, when it was worn by jazzman Donny Hathaway (1945-1979). The “Big Apple hat” was a felt hat with a wide brim, but these hats went out of style after World War II. At some time—at least by 1969—the term “Big Apple hat” was used for “Big Apple cap.”
     
“Big Apple hat” was printed in the Charlotte (NC) News on September 29, 1937. “Introducing the New Big Apple Hat” was printed in The Wisconsin State Journal (Madison, WI) on December 10, 1937.  “‘Big-apple hat,’ a wide brimmed hat” was printed in the Joplin (MO) Globe on April 9, 1946. “‘Big Apple’ hat”—the photo shows it is a cap—was printed in the Minneapolis (MN) Tribune on November 26, 1969.
   
“Big Apple Caps” was printed in the Tampa (FL) Sunday Tribune on January 30, 1938. “It’s a jitterbug cap (...) the Big Apple cap” was printed in the Knoxville (TN) News-Sentinel on May 20, 1942. Big Apple caps were advertised by many stores in 1970.

           
Wikipedia: Newsboy cap
The newsboy cap, newsie cap or baker boy hat (British) is a casual-wear cap similar in style to the flat cap.
 
It has a similar overall shape and stiff peak (visor) in front as a flat cap, but the body of the cap is rounder, made of eight pieces, fuller, and paneled with a button on top, and often with a button attaching the front to the brim (as the flat cap sometimes has).
 
Amazon.com
Jaxon Herringbone Big Apple Cap
Product description
The large profile eight-quarter cap is classic New York City streetwear. Newsboys hawked papers (‘Read all abo’t it’ ) in this cap that drips with depression era resonance. THIS IS AN OVERSIZED CAP, AND IS SIGNIFICANTLY MORE GENEROUS IN THE FULLNESS OF THE CAP. 12” diameter. 3” snap bill/visor. Made of: Wool/Poly blend. Elasticized sweatband. Supple satin lining. Colors: Grey Herringbone and Brown Herringbone. Size: 1-size fits most, fits up to size 7 3/8 comfortably. THIS CAP IS POPULAR WITH BOTH MEN AND WOMEN. PLEASE NOTE THAT THIS CAP IS OVERSIZED, AND IS SIGNIFCANTLY MORE GENEROUS IN THE FULLNESS OF THE CAP. If you want a standard 8/4 cap, this is not the cap you are looking for.
 
Levine Hats (St. Louis, MO)
LINEN BIG APPLE NEWSBOY CAP BY CAPAS
$ 59.95
Only the fullest of the full shape Big Apple Caps go here.  A big apple is a refreshing take on the 8-panel cap.  It has a very full shape, with a lot of room to adjust and style on top of your head.  If you wear a big apple cap, you’re already confident this is where you need to be.
   
Newspapers.com
29 September 1937, Charlotte (NC) News, “National Shirt And Hat Store In New Home,” pg. 10, col. 2:
On display there, one will find such fashionable articles as the Big Apple hat, novelty pleated-front shirts and Goucho shirts in different colors.
 
Newspapers.com
29 September 1937, Austin (TX) Statesman, pg. 7, col. 1 ad:
“Big Apple” Hats
The dance, “Big Apple,” is sweeping the country. The hat, “Big Apple,” is doing the same.
(Austin Shoe Store.—ed.)
 
Newspapers.com
10 December 1937, The Wisconsin State Journal (Madison, WI), pg. 3, col. 1 ad:
Smart New HATS
Introducing the New Big Apple Hat
88c
(Hill’s of Madison.—ed.)
 
Newspapers.com
30 January 1938, Tampa (FL) Sunday Tribune, pt. 1, pg. 13, col. 3 ad:
See the New
Barrel Roll Hats
and Big Apple Caps
(City Hatters.—ed.)
 
Newspapers.com
11 March 1938, The Courier-Journal (Louisville, KY), sec. 4, pg. 3, col. 5 ad:
The Season’s Newest Hat
The “Big Apple”
$5.00
The “Big Apple” hat is offered this spring for the first time, and is the leading style of the year. It features a wider brim, with welt edge, and a slightly lower gracefully tapered crown. Every hat comes with a decorative apple and leaf in the band. If you want a hat that is smartly different, see the “Big Apple” here.
(Levy Bros.—ed.)
 
Newspapers.com
1 December 1938, Jackson (TN) Sun, pg. 10, col. 5 ad:
New! New! New!
“The Big Apple” Hat $2.95
Yes, Sir. Big Apple Hats are selling in a “big” way at Kisber’s. With that new 3-inch brim…They are real dandy looking. Genuine fur water felt hats.
(Kisber’s Men’s Store.—ed.)
 
Newspapers.com
9 March 1939, Jackson (TN) Sun, pg. 8, col. 5 ad:
Men’s “Big Apple”
FELT HATS
1.79
(Kisber’s.—ed.)
 
Newspapers.com
26 October 1939, Hope (AR) Star, pg. 3, col. 1 ad:
Big Apple CAPS
All Colors 98c
(Talbot’s.—ed.)
 
Newspapers.com
18 September 1940, Chattahoochee Valley Times (Lanett, AL), pg. 7, col. 1 ad:
BIG APPLE CAPS 69c
(Jake’s Place.—ed.)
 
Newspapers.com
20 May 1942, Knoxville (TN) News-Sentinel, “Strolling” by Bert Vincent, pg. 19, col. 1:
Wears Jitterburg Cap
(...)
“It’s a jitterbug cap,” Joe (Lewis—ed.) said. “In Washington where I been everybody’s wearing it, or the Big Apple cap.”
 
Joe then said I ought to see the Big Apple. It has a top 18 inches in diameter, he said. The bill is so wide it has, in addition to one snap, two buttons, one on each side.
   
Newspapers.com
11 June 1943, Oakland (CA) Tribune, “Harlem Hailed as Zoot-Suit Birthplace, But Then Georgie Puts in Loud Claim” (AP), pg. 12, col. 2:
‘BIG APPLE’ HAT
A wide-brimmed, flat-crowned “big apple” hat, loud suede shoes, shirts with long pointed collars and screaming ties loosely knotted would be worn to complete the ensemble.
   
Newspapers.com
29 December 1945, The Times and Democrat (Orangeburg, SC), pg. 3, col. 6 ad:
Men’s Big Apple
Felt hats
50c
(Efird’s Department Store.—ed.)
 
Newspapers.com
9 April 1946, Joplin (MO) Globe, “Negro Bandit Holds Up Liquor Store,” pg. 5, col. 3:
The bandit was described by police as wearing the latest style “zoot suit” and a “big-apple hat,” a wide brimmed hat.
 
Newspapers.com
26 November 1969, Minneapolis (MN) Tribune, pg. 13, col. 3 photo caption:
EDWARDIAN SUIT, MAXI COAT
Matching ‘Big Apple’ hat
 
Newspapers.com
26 February 1970, The News-Palladium (Benton Harbor, MI), pg. 26, col. 4 ad:
MEN’S OR BOYS’
BIG APPLE CAPS
4.99
(Michigan Mart.—ed.)
 
Newspapers.com
16 April 1970, Cincinnati (OH) Enquirer, pg. 2, col. 7 ad:
THE REAL
BIG APPLE
CAP
$4.85
(Paragon Hat Shop.—ed.)
 
Newspapers.com
30 April 1970, Kokomo (IN) Tribune, pg. 5, col. 2 ad:
BIG APPLE CAPS
$6.95
(Maher’s of Kokomo.—ed.)
   
Newspapers.com
15 July 1970, Selma (AL) Times-Journal, pg. 5, col. 2 ad:
Big Apple HATS
The hat that’s “in”. Choose from several sizes and assorted colors.
3.99 4.99 5.99
(Teppers Bargain Basement.—ed.)
 
Newspapers.com
14 October 1972, Minneapolis (MN) Star, pg. 18A, col. 1 photo caption:
Big Apple caps were popularized by jazzman Donny Hathaway.
(Donny Hathaway, 1945-1979, is shown in a Wikipedia photo with this cap.—ed.)
   
Newspapers.com
12 July 1973, Daily News (New York, NY), “Night Owl” by Ernest Leogrande, pg. 87, col. 2:
Nathan Rosenberg’s hat store, 4 W. Fourth St., near NYU, next door to L’Elite Pizza Shop. Rosenberg: (...) “This is the Big Apple cap. We also call it the Applejack.”
   
Newspapers.com
24 July 1994, The Atlanta Journal/ The Atlanta Constitution (Atlanta, GA), pg. L6, col. 1:
Extra, extra! Newsboy cap is back
Jimi Hendrix wore it. So did John Lennon. Likewise Fat Albert’s pal Rudy. It’s the floppy, foppy, ridiculously large newsboy cap. And like every other fashion faux pas that lurked in the ‘60s and ‘70s, it’s back, swallowing the head of rappers, schoolkids and ambitious trendies.
 
Some are calling it the Big Apple Hat, no doubt alluding to the baggily clothed newsies on every New York street corner during the Depression. (...)—Elizabeth Lenhard
 
Newspapers.com
4 August 1996, Daily World (Opelousas, LA), advertising supplement, pg. ?:
Sale $6
Reg. $8. The Big Apple hat. Available in denim and denim with suede.
(JCPenney.—ed.)
 
Newspapers.com
24 October 2007, Detroit (MI) Free Press, pg. 8A, col. 5 ad:
Big Apple Caps
(City Hatter.—ed.)
 
Twitter
James Brown Jr
@JamesBrownJr9
Replying to @RollingStone
@RollingStone @justinbieber must bring back the big Apple hats
8:56 PM · Nov 20, 2015 from Dundalk, MD·Twitter for Android
 
Gentleman’s Gazette
Flat Cap & Newspaper Boy Hat Style Guide
October 20, 2017/34 Comments/in Accessories, Hats, Outfits, Videos, Wardrobe, Clothing and Accessories /by Sven Raphael Schneider
Flat caps have had a renaissance in recent years, and for good reason. Not only are they warm in the winter, they are classicly stylish and nearly any well-dressed man of any age can pull one off.
(...)
Depending on the location, the flat cap can be referred to by more than 20 monikers: cabbie, paddy, Gatsby, dai, longshoreman’s, scally, Wigens, ivy, derby, Jeff, duffer, duckbill, driving, bicycle, Irish, or a crook cap. It is also known as a sixpence, bunnet, cheese-cutter, or a Vergon or Joao’s hat.