For the Khrushchev years, you must read William Taubman's Khrushchev: The Man and His Era (2003), which is the definitive work on the subject. I also recommend A. A. Fursenko's Khrushchev’s Cold War: The Inside Story of an American Adversary (2006). Vladislav Zubok and Constantine Pleshakov's Inside the Kremlin’s Cold War: From Stalin to Khrushchev (1996) is a great overview of the decision-making and motivation of the Soviet leadership during this period, and Zubok's Zhivago’s Children: The Last Russian Intelligentsia (2011) explores the experience of Soviet elites.

For something more focused on popular reactions, you could check out Rob Hornsby's Protest, Reform and Repression in Khrushchev's Soviet Union (2013).

And of course, since you don't want to ignore the global environment, I recommend Odd Arne Westad's The Global Cold War: Third World Interventions and the Making of Our Times. (2005), which is not about the Soviet Union per-se but has excellent chapters on the Cuban Missile Crisis and other major moments of intervention and conflict that shaped Soviet history during this period.

For something that connects the previous era with the era you're interested in, I can't recommend Zubok's A Failed Empire: The Soviet Union in the Cold War From Stalin to Gorbachev (2008) enough. It's a great overview.