Wiktionary
adv. (context US idiomatic English) After reconsidering; on further consideration. alt. (context US idiomatic English) After reconsidering; on further consideration.
Wikipedia
"On Second Thought" the title of a song written and recorded by American country music artist Eddie Rabbitt. It was released in November 1989 as the first single from the album Jersey Boy. The song topped both the United States and Canadian country charts for two weeks in 1990. It was the final No. 1 song for Rabbitt, who died from lung cancer in 1998.
The song marked a return to traditional country for Rabbitt since his emergence as a crossover success in the late 1970s and early 1980s. It had been described as a "country shuffle." A re-recorded version of the song was also included on his 1997 album Beatin' the Odds.
Usage examples of "on second thought".
He understood it and started to answer, but then, on second thought, he decided not to reveal his knowledge of the language.
Though on second thought our records reveal that he is even stupider than you.
Yes, on second thought, Quintiform was decent enough he conned himself into thinking.