The International Workers Order (IWO) was a Communist Party-affiliated organization founded in 1930 and disbanded in 1954 as the result of legal action undertaken by the state of New York in 1951. At its height in the years immediately following World War II, the IWO had almost 200,000 members.

Obviously, that doesn't account for all American Communists, but it gives a general idea.

Edit: the US population around that time was about 150 million.

Klehr, Harvey (1984). The Heyday of American Communism: The Depression Decade. Basic Books. pp. 3–5

More info on the IWO:

The U.S. Attorney General placed the IWO on its list of subversive organizations in 1947. The New York State Insurance Department contended that, since the IWO was engaged in political activity, which was prohibited to insurance organizations, it placed its members' interests in jeopardy, leading ultimately to its liquidation.

Arthur J. Sabin, Red Scare in Court: New York versus the International Workers Order. Philadelphia: University of Pennsylvania Press, 1993

A note on my answer: I make a distinction between communism and socialism, and communist-leaning organizations and those actually communist.

The peak of the Communist Party of the United States came in 1938, when an estimate put its rolls at 75,000 members. This peak coincided with the end of the Great Depression, which pushed Americans to political extremes (right and left) in the search for solutions to the economic problems facing the country.

It also came during the (Franklin) Roosevelt administration, which, if not fully open to communists, did not shun their support. In 1935, the Comintern spread the idea of an international "popular front" against fascism, which led the CPUSA to support the New Deal and the Democratic Party in general. As it was no longer directly promoting revolution and violent overthrow (at least in the minds of many Americans), it became more acceptable to be communist.

Furthermore, the apparent (to outside eyes) Soviet success in staving off the Great Depression and massively industrializing made it appear as if communism was succeeding in the Soviet Union. This led thousands of Americans to leave the United States for the Soviet Union (see Tim Tzouliadis' The Forsaken: An American Tragedy in Stalin's Russia). Although small in number, their travels are an important indicator to popular thought at the time.

The downward slide of the CPUSA, in hindsight, began with the signing of the Molotov-Ribbentrop Pact. This abrupt reversal of the anti-fascist stance disillusioned many communists outside the Soviet Union. While WWII propaganda did much to raise the stature of the Soviet Union during wartime, this rapidly decayed after the war with the start of the second Red Scare. By the 1950s, according to some estimates, the CPUSA had more FBI informants and agents on its rolls than it did real members.

Nowhere in the United States did communism have greater influence at its peak than in Washington state. James Gregory of the University of Washington has a pretty good project going on right now about it, and I'd encourage you to check it out if you have the time. As Postmaster General James Farley joked in 1936: "There are 47 states in the union, and the Soviet of Washington.

You oughta give "Witness," by Whittaker Chambers, a read. It's a (pretty long) tale of his journey into Communism, his conversion to Christianity resulting in his departure from the Party, and his role in getting Alger Hiss (a friend, a fellow traveler, and a fairly high-up State Department employee) convicted of perjury.

It's a fascinating book. My mother gave it to me, and I texted her one day that I was having trouble keeping all the people straight. She answered, "I know, right? Commies everywhere!" And there were. They were in all levels of the federal government. Chambers goes into depth about the mindset that made Communism seem like the solution to the problems that the world was facing at the time.

It does get pretty philosophical, but that sounds like it might be exactly what you're looking for.