Yeah, the late 60s and early 70s is also where it is at for me! And I must admit that I am a country nerd and think that Elvis more stripped-down recordings in the genre are just incredible – he embodied the high and lonesome sound like few others have done.
I guess that another reason that he was such a powerful singer is that he picked some of his saddest songs based on how he felt deep down inside and that he truly believed the sentimental lyrics that he was singing, no matter how cliché they were – “Always on My Mind” and “Tomorrow Never Comes” are two examples that never fail to put a tear in my eye. He was not just singing from his heart, as they say – it is like it was primal.
And great question about recommendations about essential stuff – really appeals to my nerdiest side (one of my worst ones, if you ask my wife)! 
For starers, I cannot recommend Peter Guralnick’s two part biography enough – Last Train to Memphis and Careless Love. It is easily the best piece of musical journalism that I have ever encountered. Those books are just insanely good. And if it were not for the fact that it is non-fiction, they could be a contender for The Great American Novel.
When it comes to the music, when I started to make my way down the rabbit hole in the mid-90s, the Essential 50s, 60s and 70s Masters boxes were the bomb and I still love those (the 70s box looks so damn good). If you want his movie songs there is also a 2 cd companion piece, Command Performances (love the title more than the songs). The stellar single disc 1956 nicely covers that legendary year. There is a collector’s edition in hard cover that is a bargain – always go for the “Elvis porn”, I say. I also remember buying a disc called Tomorrow Is a Long time that included some great 60s songs like “Guitar Man” that were not in the boxes.
When it comes to the non-over dubbed stuff (my hard drug), I believe that the more recent boxes rounds up most of the stuff from the 70s: Elvis at Stax, From Elvis in Nashville and Elvis Back in Nashville. But when we get to the Memphis sessions, things get more complex. The ancient but excellent 2 cd Suspicious Minds (The Memphis 1969 Anthology) – curated by Guralinick! – covers pretty much all the must-have official recordings, but there is more stuff without overdubbs to be had on releases like the 2 cd From Elvis at American Sound Studio. Some of the songs I listen to seem to be from releases called Unedited Masters and Alternate Masters, but I do not know if those are official (got parts of my physical collection in the attic) and to what extent there are overlaps. I simply cannot keep track of all the different takes of the songs – there are just too many.
An anecdote about the Suspicious Minds (The Memphis 1969 Anthology). There was this really good local record store back in the days. When a friend of mine asked if he could listen to that album on the headphones, the dude in the store just looked at him and said something like: “Now, son. Lets make things easy for the both of us. Fork up the 15 dollars and I will put the album in a bag for you. It is one of the best albums that you will ever come across”.
One can also find various gems from the late 60s and 70s scattered all over – often expensive as hell – boxes and releases, many of them in the Follow That Dream (FTD) series. I am not up to date since my nerdiest period was before the last two Nashville boxes came out, but looking through my playlist I found, for example, the take I like the most of “Help Me Make It Through the Night” is on the Elvis Now FTD edition and the version of “Burning Love” that I play is taken from the Platinum – Life in Music box. (If I remember correctly, I simply bought those from iTunes.) Again, I am not completely sure about how much the recent 70s boxes round up – maybe those are included.
