Glumac started his career playing at youth level for his hometown club GOŠK Dubrovnik. At the age of 15 he joined Hajduk Split where he eventually signed a professional five-year contract. In the second part of the 2009–10 season, he was loaned to Zadar to help the team avoid relegation. He was a first-team regular and played in 13 matches that season. He also remained at Zadar the following season and scored his first goal in Prva HNL in a 3–2 defeat to RNK Split. In the 2011–12 season, Glumac returned to Hajduk and made his debut for the first team on 21 October 2011 as a late substitute in the 3–0 win against Lučko. A month later, manager Krasimir Balakov announced that Glumac was no longer a part of the first team. On 9 January 2012, it was announced that Glumac was transferred to RNK Split in exchange for Goran Milović, signing a two-and-a-half-year contract.