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Reagan, to my knowledge (I was quite young), ran within the Republican Party. He wasn’t from the establishment wing. He belonged to the Goldwater anti establishment wing. I don’t think he ever ran a true third party campaign (as in outside the two main parties).

I think this is what mark levin was talking about, Reagan taking on a sitting President..

from Wiki:

The 1976 National Convention of the Republican Party of the United States met at Kemper Arena in Kansas City, Missouri, from August 16 to August 19, 1976. The convention nominated incumbent Gerald Ford for President, but only after narrowly defeating a strong challenge from former California governor Ronald Reagan. The convention also nominated Kansas Senator Robert J. Dole for Vice President. The keynote address was delivered by Tennessee Senator Howard Baker.

Although Ford had won more primary delegates than Reagan, as well as plurality in popular vote, he did not have enough to secure the nomination, and as the convention opened both candidates were seen as having a chance to win. Because of this, both Ford and Reagan arrived in Kansas City before the convention opened to woo the remaining uncommitted delegates in an effort to secure the nomination.

Reagan benefited from his highly committed delegates, notably "Reagan's Raiders" of the Texas delegation. They and other conservative Western and Southern delegates particularly faulted the Ford Administration's foreign policy of détente towards the Soviet Union, criticizing his signing of the Helsinki Accords and indirectly blaming him for the April 1975 Fall of Saigon.

The pro-Reagan Texas delegates worked hard to persuade delegates from other states to support Reagan. Ford, meanwhile, used all of the perks and patronage of the Presidency to win over wavering delegates, including trips aboard Air Force One and personal meetings with the President himself.

As then President Richard M Nixon was the previous nominee in 1972, the nomination of President Ford in 1976 meant that a major party had nominated the sitting President for the second time in a row, the first time this has happened since the Democratic Party renominated Franklin D Roosevelt for a third term in 1940.