Montauk, NY early 1900's  

The view of the depot is looking northeast and shows the relationship between the 2nd (1927) and 3rd (1942) depot buildings as well as the short distance the LIRR terminal was moved south in 1942 when the U. S. Navy took over the LIRR’s former property.

The shot with the cattle round-up is c. 1955 and you can see that the 1927-era depot in the left background has already had its long covered platforms removed.  The old, wooden freight house is across from the 1942-era depot.
Archive: Art  Huneke,  Info: Dave Keller

      
PRR K4s with RPO car and heavyweight parlor cars arrives eastbound at station – Montauk, NY – c. 1932

(Per Art Huneke:  The configuration of the power reverse and the air reservoir indicate that the engine in question is a K4s and not a G5s.)

K2s, K3s and K4s locomotives began replacing G5s locomotives making Montauk runs after the heavier “K4 bridge” (as it was nicknamed) was installed over Shinnecock Canal and placed in service in June, 1931.

This depot, the 2nd one at Montauk, was constructed in 1927 parallel to the end-of-tracks, bumpers of which can be seen in front of the depot.  Also visible is the keystone station sign hanging from the eaves.  This was all removed in 1942 and moved slightly south, with a newer (3rd) depot constructed parallel to the end-of-track when the U. S. Navy took over the LIRR property for a torpedo base on Fort Pond Bay.  This 1927-era depot (rebuilt and missing one covered platform) still stands on the grounds of a condominium complex which has been built on the former U. S. Navy site.  Dave Keller archive/research

LIRR #109 H10s passenger train WB Montauk c.1947 Photo/Archive: Art Huneke

Montauk, NY c.1910
Archive: Dave Keller 

Montauk Beach poster featuring the Montauk Manor Hotel c. 1927 Archive: Steve Lynch

Montauk c.1905
There is no electricity as is evidenced by the Dietz kerosene platform lamp on the station wall past the door. The wire you see is the telegraph line. Info: Dave Keller, Archive: Dave Morrison

Distant shot of the depot in this photo, you can make out the vestiges of the overhanging eaves at the former 1st level roof as the structure continues above and out back. Archive/Info: Dave Keller

2-story photo shot above shows a length accommodating 3 equally-spaced windows deep and this is 3 equally-spaced (almost) windows deep. 

Art Huneke suggests this rear area was added in 1898 which is most likely as the Army Camp created  quite a bit of growth out there around that time. Info: Dave Keller, Archive: Art Huneke

About 50 years later.

Montauk Station History
Info: Dave Keller

BUILT: 1895, REMODELED TO 2ND STOREY: c. 1907, RAZED: 1927

2ND DEPOT OPENED PERPENDICULAR
TO END OF TRACK: 6/1/27, 
CLOSED: 2/42 WITH TRANSFER OF RAILROAD PROPERTY TO U.S. NAVY.

3RD DEPOT OPENED JUST SOUTH OF PRIOR LOCATION: SPRING /1942 
YEAR-ROUND AGENCY CLOSED: 
01/21/72. OPEN SUMMERS ONLY THEREAFTER. CLOSED 
PERMANENTLY: 19 ? . IN USE 
AS STORE.

4TH DEPOT WITH HI-LEVEL 
PLATFORMS OPENED: 2000-01. 

Jim Osborne began his LIRR career in 1921 and the photo he took of Montauk would have been shot anywhere between 1921 and 1927; prior to when the structure came down for the 2nd depot perpendicular to the end of track.
Info: Dave Keller, Archive: Art Huneke

Leased PRR K4s #719 and train is laying up eastbound at the bumper block of the 1927-era station facilities at Montauk, NY c. 1937. Notice the crossover switches slightly behind the tender, allowing the train to be pushed slightly west after all passengers have gotten off, engine and tender cut off the train, then over the crossover switches onto the south passenger track, heading west to access the wye for turning for the trip back. These facilities were taken over by the U. S. Navy in 1942 to build a torpedo-boat base and the LIRR facilites were moved slightly south to their present-day location with all new tracks, new yard and new depot. One major change was that the similar style depot building was constructed parallel to the tracks as opposed to this layout stubbing out perpendicular to the structure. Dave Keller archive

I'll go on the outside edge and say it was shot just before the bridge opened as it is not shiny and new-looking . . . .as though it's just been delivered. 
Archive/Info: Dave Keller

FM2006 at Montauk
"Handing up Train Orders?"
The "crew" posing for a moment

FM H16-44 #’s 1506, 1504, doubleheading train eastbound at station platform  Montauk, NY c. 1958. Conductor in blue uniform on platform.  Engineer Richard L’Hommedieu in cab of  lead unit. Collection: James Caramore 
Archive: Dave Keller

LIRR #2004 CPA20-5 Backing Baggage, RPO/REA cars onto Wye Bunkhouse at left View SW 03/19/1955 Archive: Dave Keller

July 29, 1968: Bob Emery backs the 12 cars (note six parlors) of train #5, The Cannonball, into Montauk Station. It's barely daybreak and Bob's primary piece of flagging equipment, a white handkerchief, works just fine. The Mattituck is the rear car of #5. Bob and I frequently worked parlor trains together. 
Photo: Richard F. Makse

South leg of wye, view N from Industrial Rd. 03/21/2008
Photo: Mike McDermet

LIRR C3 coach #5013 Montauk 
View SE 08/21/2011