He returned to Independiente before moving on to Boca Juniors (1936), Club Atlético Platense (1937–38), and Almagro (1939–40); he also played for Flamengo in Brazil (1939; 1940), Peñarol in Uruguay (1941–42), and Santiago National in Chile (1943).

Over the next 12 years, he played 13 times for Argentina and scored 3 goals, winning the 1927 Copa América and the silver medal at the 1928 Summer Olympics in Amsterdam, Netherlands.

[6] Orsi's career is strange by modern standards, however, in that he played for Italy as well as Argentina, allowing him to gain 35 caps and score 13 goals for his second country between December 1, 1929, and March 24, 1935.

A prolific goalscorer, he was an accurate finisher, both with his head and his feet, and he excelled in the air and acrobatically; because of this he was also capable of playing as a striker, and on the right flank, a position in which he was able to make diagonal attacking runs or cut into the centre to shoot with his stronger foot.

Nicknamed "Mumo", despite his lack of shooting power and physical strength, he was an extremely quick player, with excellent technical ability, who was renowned for his dribbling skills and his use of feints to beat opponents.