Catherine Merridale’s Ivan’s War is the most accessible social history of the Red Army, though it leans a mite heavily into the gloom and misery aspects of the war perhaps (which to be fair, there was plenty of gloom and misery to go around.)
Roger Reese’s Stalin’s Reluctant Soldiers is more academic, which is both good and bad depending on your tastes and familiarity with the subject.
Brandon Schecter’s PhD thesis, “Government Issue,” is a very in depth look at the material culture of the Red Army; one of the best pieces of recent Red Army scholarship in my opinion. It’s available online for free if you google it, and it was turned into a book last year which I have not read yet (The Stuff of Soldiers).
Alexander Hill’s The Red Army and the Second World War is pretty comprehensive, though it is a general history rather than pure social history so finding particular tidbits might be hard.
You can’t talk about the Red Army and not talk about the women who served; Markwick and Cardona’s Soviet Women on the Frontline in the Second World War is my go-to, though Anna Krylova’s Soviet Women in Combat also comes highly recommended. Svetlana Alexeivich’s Unwomanly Face of War is oft recommended but not without flaws; I would keep a couple of the more critical reviews at hand while reading it for the first time.
Memoirs can be quite hit or miss because of the nature of censorship during and after the war. Boris Gorbachevsky’s memoirThrough The Maelstrom: A Red Army Soldier’s War on the Eastern Front has been on my radar recently, but I have not read it yet.
Kenneth Slepyan’s Stalin’s Guerrillas is a great social history of the partisan movement, as well.