I don't think the republicans were ever a third party, when they formed in 1854, the Whigs were only 22 years old.
The republicans simply replaced the Whigs, two years after they first met in 1854, John Fremont came in second in the 1856 presidential race, the “know Nothings” “third party” came in a distant third with 8 electoral votes. The Whigs were pre civil war America, the party only lasted 22 years itself (1833 to 1856). The Republicans held their first convention on July 6, 1854 and four months later held 19% of the Congress and 25% of the Senate, six years later they held the majority in the Congress, the majority in the Senate, and they had the Presidency. The Republicans were never really a third party at all.
Your original post said “The Whig party became the Republican party,” indicating you believe the GOP grew out of the Whig party. That’s not true, and that’s the point I was trying to make.
“The republicans simply replaced the Whigs...”
I would agree with that. But, the important point is that the Republican party was not just a renamed Whig party.