Curious of the origins of this song, which I only know as part of Elvis' I Got A Woman/Amen medley, I looked what could be found about it on the internet.
My starting point, as usual for these kind of threads, is the wonderful site http://davidneale.eu/elvis/originals/list1.html#S1124 from David Neale, who did a lot of fantastic research on this. He writes:
Amen
Recorded by Elvis on Wednesday, 20 March 1974; Concert
Written by: Traditional
Originally recorded by Woody Herman in 1942
Hear Elvis's version on: Recorded Live On Stage In Memphis
RCA also recorded "Amen" (sung together with "I Got A Woman") on April 9, 1972 (Coliseum, Hampton Roads, Virginia). Though not officially released, the recording can be heard on "Standing Room Only."
An even earlier recording, made in Boston on 10 November 1971, can be heard on the unofficial release "The Power of Shazam."
The song was used in a 1940 film called "Maryland," but the first released recorded version seems to have been as a sort of swing parody by Woody Herman, otherwise referred to as "Hey Man!" from the 1942 film "What's Cookin'?" (Decca 18346).
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The link between the Herman's version and the more traditional "Amen" is, however, tenuous, so that perhaps a more likely contender for original version is the 1950 recording, released on King 4372, by Rev. J.B. Crocker, titled "Sermon, Hallelujah Amen."
Incidentally, the song featured heavily in the 1963 film "Lillies Of The Field," starring Sidney Poitier and earning him the first ever Best Actor Of The Year Oscar awarded to a black player.
There is one sentence in Woody Herman's version that is quite remarkable. We'll hear it at 2:18 in the YouTube clip that was quoted. It sounds like this was the inspiration for the first verse of Elvis' Hard Headed Woman: "A hard headed woman, a soft hearted man, been the cause of trouble ever since the world began". In Amen (1942) we hear Herman sing: "A two faced woman and a jealous man, cause the trouble since the world began".

Woody Herman and Band at Brooklyn Armed Guard Center 9 August 1944
Though the original Decca label tells us: "From Universal Picture "What's Cookin'?", the version we hear in the picture itself is quite different. "What's Cookin'" was a showcase for Woody's band and also featured the Andrews Sisters, Jane Frazee, Donald O'Connor, and Gloria Jean. This faux spiritual from the film turned out to be a hot record seller for both the Herman and Abe Lyman bands. What follows is the original movie clip from the now-obscure Universal musical film.

The clip is from Castle Films' home-movie reel "Song Festival," which presents three musical numbers from the 1942 Universal Pictures film "What's Cookin'?" This rousing number is "Amen (Are You Happy?)" featuring the Andrews Sisters, Jane Frazee, and bandleader Woody Herman, assisted by the Jivin' Jacks and Jills (Universal's pool of teenage dancing and singing talent). Keep your eyes open for a young Donald O'Connor, an as-yet-unbilled member of the Jacks and Jills, and for Gloria Jean and Peggy Ryan. This trio would appear together in three more B musicals for Universal.
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As David Neale points out: "The link between the Herman's version and the more traditional "Amen" is, however, tenuous, so that perhaps a more likely contender for original version is the 1950 recording, released on King 4372, by Rev. J.B. Crocker, titled "Sermon, Hallelujah Amen." It also can be found as a bonus track on Document Records DOCD-5524 "Elder Charles Beck - 1946-1956 In Chronical Order".
Here it is:
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And this is the way it was presented in Ralph Nelson's "Lillies Of The Field" in 1963. (It starts with an instrumental version, around 2:00 the first sung version starts. Unfortunately the talking done is a French dubbing). Sidney Poitier won an Academy Award (Oscar) for his acting in this film. The words to the song "Amen" in this film are written by Jester Hairston, as is the arrangement.
First appearance in the film (at the film's beginning).
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Later appearance of the song in the film (at the film's ending).
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Jester Joseph Hairston (July 9, 1901 – January 18, 2000) was an American composer, songwriter, arranger, choral conductor, and actor. He was regarded as a leading expert on Negro spirituals and choral music. He arranged "Amen" for "Lilies of the Field" and is considered by some as the composer of Amen. For the song with his words and arrangement was a 1963 hit for The Impressions:
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The song went to number one on Cashbox Magazine's R&B chart for three weeks and reached #7 on the Billboard Hot 100 singles chart in 1964. When we listen to this version and take a look at the credits given on Elvis' albums, we can say that Elvis' version was heavily inspired by this arrangement and this specific version of Amen. Though he might have known the traditional much much longer.
A new version was released by The Impressions in 1969 under the title "Amen (1970)", reaching #44 on the Billboard Best Selling Soul Singles chart in January 1970. This version was composed and arranged by Jazz bassist, producer & arranger John William Pate.

Here is this 1970 version as sung by Tommy Ellison & Five Singing Stars:
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As far as I know, Elvis started to sing Amen as part of the I Got A Woman/Amen medley somewhere around November 1971. An early version can be heard on this FTD release:

Why Elvis combined Amen with I Got A Woman? I have no idea. Maybe to show that spirituals and rock'n'roll are closely interrelated? To start his show with a spiritual uplift?
The version that is most known among Elvis fans, can be found on:

Personally I would have liked it more if Elvis sung it separately to raise good spirit (if that was this aim). Perhaps together with Oh, Happy Day in a spiritual medley or section of the show.
To close this OP, a few other covers of the song:
1. Prior to the Impressions version (December 1960), Marv Meredith's "Salvation Rock" (essentially a version of "Amen") reached the Music Vendor national Top 100.
2. Also a minor chart hit for Lloyd Price and Erma Franklin in 1964 (Bubbling Under Hot 100 Singles at #124)
3. In 1968, Otis Redding had a posthumous hit with his version of the song, reaching #15 on the R&B chart.
4. In 1969, The Winstons recorded an instrumental funk cover, which included the infamous Amen Break.
SOURCES
http://davidneale.eu/elvis/originals/list1.html#S1124
https://www.youtube.com/watch?v=99srNEZYDIs
https://www.youtube.com/watch?v=cHTu6coyUog
https://www.youtube.com/watch?v=nRnaG4-HJmc
https://www.youtube.com/watch?v=kH1d9i6-9L8&spfreload=10
https://en.wikipedia.org/wiki/Jester_Hairston
https://en.wikipedia.org/wiki/Amen_(gospel_song)
https://www.discogs.com/Impressions-Amen-1970-Wherever-She-Leadeth-Me/release/2932970