After two seasons deputising for Hans Galjé, and a loan to fellow league side HFC Haarlem, he became the starter for the 1985–86 campaign. Newly appointed manager Johan Cruyff believed that Menzo was one of the first goalkeepers who could also make his mark as a field player.
In the 1994 summer, Menzo signed with PSV Eindhoven, where he backed up Ronald Waterreus for two seasons. The 33-year old managed to revive his career in Belgium with Lierse SK, which he helped win one league and one cup, eventually amassing nearly 100 official appearances. For a brief period of time, he also played in France for FC Girondins de Bordeaux, arriving in August 1997 to replace Gilbert Bodart. However, having lost his place to Ulrich Ramé (who would be the club's first-choice for more than one decade), he returned to Lierse, in January 1998.
In the summer of 2001, after a second spell at Ajax, Menzo joined amateur club AGOVV Apeldoorn, helping it to the amateur title. He retired from football at the end of the season. When coach Peter Bosz left in June 2005, he became AGOVV's manager, remaining in the position for only one season: AGOVV became a professional club in the second division, but he did not have the qualifications to exercise in that category, subsequently moving to Amsterdam-based amateur club Amsterdamsche FC.
In February 2005, Menzo received the necessary diploma to coach professional clubs. That summer, he returned as head manager of AGOVV. A year later, he joined FC Volendam, lasting two seasons (incomplete) and signing with Cambuur Leeuwarden, which he led to the second position in the second level in 2009–10, even though the team ultimately failed in the playoffs.
In October 2010, Menzo resigned from his position at Cambuur in order to join Vitesse as assistant to new head coach Albert Ferrer. In May 2013, he became the head coach of Lierse S.K. .
Menzo gained six caps for Holland, the first arriving in 1989. He then spent nearly three years without any further appearances, but was summoned for the squads present at both the 1990 FIFA World Cup and UEFA Euro 1992, as third-choice.
After legendary Hans van Breukelen retired from international play following the latter competition, Menzo was named the starter for the qualification stages of the 1994 World Cup; following two unassuming performances, the 1–2 loss in Norway and a 2–2 home draw against Poland, he was benched, and ultimately did not even made the squad at all (Van der Sar would be the third goalkeeper in the list).
In the summer of 2004, after former Ajax teammate Marco van Basten became head coach of the national team, Menzo was named its goalkeeping coach, remained there for two years.