[2] After his ordination, Muench joined the faculty of St. John Vianney Preparatory School in New Orleans He worked there until 1977, serving as religion teacher, liturgy director, guidance counselor, and eventually rector.

In 1983, he was named director of the Pope John XXIII House for Vocation Discernment and executive assistant to Archbishop Hannan.

[1] He was consecrated on June 29, 1990, at Saint Louis, Cathedral in New Orleans by Archbishop Francis Schulte, with Hannan and then Bishop John Favalora serving as co-consecrators.

[3] On January 5, 1996, John Paul II appointed Muench as the ninth bishop of the Diocese of Covington.

[1] Muench served as a board member of St. Elizabeth Medical Center in Northern Kentucky, chancellor of Thomas More College in Crestview Hills, Kentucky, and a member of the Committee on Priestly Formation within the United States Conference of Catholic Bishops (USCCB).

He also initiated renovations of the Cathedral Basilica of the Assumption[1] On December 15, 2001, John Paul II appointed Muench as the fifth bishop of the Diocese of Baton Rouge .