Hong, who made 136 appearances for South Korea and captained them to the semi-finals of the 2002 World Cup on home soil, had signed a contract until the 2027 Asian Cup, Yonhap news agency reported.
He last took charge of the side for 19 games between 2013 and 2014, leaving the role after their group stage exit at the 2014 World Cup.
Hong's return to the national team comes after he led Ulsan HD to back-to-back K League titles in the past two seasons.
He has been tasked with taking South Korea to an 11th straight World Cup, with the team having progressed to the third round of qualifiers for the 2026 tournament in Canada, Mexico and the United States.
Hong replaces German coach Klinsmann, who left following the team's shock semi-final defeat by Jordan at the Asian Cup.
Hwang Sun-hong and Kim Do-hoon served as interim coaches while the KFA searched for a permanent replacement.
According to reports in South Korean media, Canada coach Jesse Marsch, ex-Norwich City manager David Wagner and former Greece coach Gus Poyet were among the candidates considered for the role.