(This is partially adapted off an earlier answer of mine.)
Your question about "other ultra-conservative groups" is perhaps a touch too broad; you could include, for example, the Free Society of Teutonia founded in 1924, later renamed the Friends of the Hitler Movement, or the Defenders of State Sovereignty and Individual Liberties from 1955 (a Virginia group dedicated to school segregation). I'm guessing you're looking for continuity with the John Birch Society's main attributes: anti-communism to the point of extreme conspiracy theories. (Before the JBS was even officially formed in 1958 the founder Robert Welch was forming a "Black Book" making the argument that Eisenhower was a secret Communist.)
In that case, I can direct you to the Minute Women. They were founded in Connecticut (1949) by Suzanne Silvercruys Stevenson, but their biggest and most active chapter was eventually in Houston, Texas.
They were essentially Communism vigilantes.
They worked by "telephone chain", so that one member would call five, and then each of those would call five more, such that they could quickly get hundreds of people involved in an effort; they stopped Dr. Rufus Clement (president of Atlanta University) from lecturing at a Houston church for being "too controversial"; they stopped a Quaker meeting because of supposed links to Alger Hiss to Quakerism. They planted observers in University of Houston classrooms to screen for controversial material.
They got schools to ban a UN essay contest; they printed a false report that "troops flying the United Nations flag once took over several American cities in a surprise move, throwing the mayors in jail and locking up the police chiefs." (This was a distorted riff on what was actually an Army exercise, done in collaboration with the government.)
This was all in the early 50s; they lasted all throughout the Communist paranoia phase up to the 60s. (Perhaps most spectacularly, in 1956 they were part of a protest against the Alaska Mental Health Bill, claiming it was intended make concentration camps to house political prisoners, "our own version of the Siberia slave camps run by the Russian government.")
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For the JBS as I mentioned, their officially founding by Welch was in 1958, where they were decried by mainstream conservatives (like Buckley) as "fringe", and essentially picked up where McCarthy left off in accusing everyone of being communists. In 1961 Welch claimed "50 to 70 percent" of the US government was controlled by Communists.
In addition to anti-communism, they were (are, they're still around) very much against the United Nations for the same Fear of World Government + Racism reasons all the rest of the far right were.
One of their allies was General Edwin Anderson Walker, US Army officer and veteran of both World War II and the Korean War. He picked up the same views as the John Birch folks (I'm not sure the sequence if he met them first or vice versa) and started to get in trouble trying to indoctrinate his troops. He tried to resign in 1959; Eisenhower refused, and he got reassigned. He tried to resign again in 1961 with Kennedy; this resignation was accepted. By then he had gone on record as essentially calling Truman a Communist.
He later gave a speech in Dallas, claiming he could no longer serve in uniform and collaborate with "the release of United States sovereignty to the United Nations" and the "subversion of national interests by one-worlders." (These are direct JBS talking points.)
He ran for governor in 1962 (coming in dead last amongst the nominees) but continued with bizarre and conspiratorial anti-Communists tirades. In 1963, Lee Harvey Oswald (who considered Walker to be the leader of a fascist group and compared him to Hitler, and had attended one of his meetings) photographed Walker at his home in March, and then attempted to shoot him through a window in April. The bullet hit the window frame; Walker was injured by fragments. There were no suspects, although Oswald was realized as a suspect after the Kennedy assassination.
Later that same year Adalai Stevenson (ambassador to the UN) came to speak in Dallas on October 24. Walker had a rally the day before and invited far-right groups (including the JBS) to buy tickets for Stevenson's talk.
Stevenson was met by a jeering crowd (and allegedly, a banner reading "US out of the UN") and had to cut his speech early.
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So: the JBS could form conspiracy theories, make Joseph McCarthy look sedate when it came to accusations of Communism, and coordinate attack stunts. But how much political power did they really have?
They formed a large enough voting block that the mainstream Republicans that wanted to distance themselves from Welch had to do so in a way that wasn't alienating the members, which got to an estimated 60 to 100 thousand people.
Goldwater in particular was claimed as one of theirs (he supposedly had a copy of the secret Black Book from the early days). Goldwater said Welch was an extremist but the JBS itself was full of fine people. He was funded partially by the JBS and was benefit to their volunteer activities.
Goldwater won the Republican nomination in 1964 (that'd be a whole different post, but the South was reacting poorly to civil rights reforms and the most likely opposition candidate of Rockefeller was too moderate for them) but his momentum started to unravel right at his acceptance speech. He declared
I would remind you that extremism in the defense of liberty is no vice! And let me remind you also that moderation in the pursuit of justice is no virtue!
basically channeling JBS wholesale. Things started to go downhill from there, where the moderates of the Republican party kept their distance and the specter of total nuclear destruction put fear in the people of the United States. The legendary "Daisy" ad aired in 1964 only once (you can watch it here via the Library of Congress) but once was enough -- it showed a girl counting flower petals in a field before getting annihilated by a mushroom blast.
Goldwater lost utterly to Johnson (witness: the electoral map). The JBS still maintained some political power through sheer numbers, but started to focus on more unhinged conspiracy theories like all police forces in the country uniting to form a Communism "super force", or promoting laetrile as a cancer cure. (It is not even remotely a cancer cure and can be lethal when consumed.) They have had a resurgence in power in recent times but far too close to the 20-year cutoff of this sub to discuss in detail; suffice it to say the society itself is still around and their methods of using conspiracy theories to build fanaticism have been appropriated by other groups.
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Conner, C. (2013). Wrapped in the Flag: A Personal History of America's Radical Right. Beacon Press.
Mallon, T. (4 Jan. 2016). The John Birch Society and the rise of the radical right. The New Yorker.
Mulloy, D. (2014). The World of the John Birch Society: Conspiracy, Conservatism, and the Cold War. United States: Vanderbilt University Press.
Nickerson, M. M. (2004). The Lunatic Fringe Strikes Back: Conservative Opposition to the Alaska Mental Health Bill of 1956. The Politics of Healing: Histories of Alternative Medicine in Twentieth-Century North America, 117-130. Routledge.
Perlstein, R. (2009). Before the Storm: Barry Goldwater and the Unmaking of the American Consensus. United States: PublicAffairs.
Smith, R. (2014). On His Own Terms: A Life of Nelson Rockefeller. Random House.
Toy, E. (2004). The Right Side of the 1960s: The Origins of the John Birch Society in the Pacific Northwest. Oregon Historical Quarterly, 105(2), 260-283.
