With OB, Høgh continued his consistent goalkeeping, and guided the team to another Danish championship in 1989, as well as Danish Cup triumphs in 1991 and 1993. He won the "Best Goalkeeper of the Year" award a further four times, setting a record five award wins in 1994. Høgh was a part of the Danish national team at the 1995 King Fahd Cup, where he was Denmark's stating goalkeeper in the absence of Peter Schmeichel. In the second game of the tournament, Høgh suffered an injury, and was replaced by Mogens Krogh, who tended the goal as Denmark won the tournament.