On 17 April 2010, Bayer 04 Leverkusen agreed terms with Croatian club Rijeka for the signing of the 19-year-old midfielder on a five-year deal.

In January 2013, the loan at MSV Duisburg was terminated and Pamić returned to Croatia to join Dinamo Zagreb.

On 9 January 2019, Pamić signed a contract with Premier League of Bosnia and Herzegovina club NK Široki Brijeg.

[11] Pamić made his league debut for Široki Brijeg on 7 April 2019, in a 1–0 home win against NK GOŠK Gabela.

[12] In the 2018–19 season, Pamić and Široki Brijeg finished as Bosnian Cup runner-ups after losing to FK Sarajevo in the final.

Zvonko's older brother was Alen Pamić, a professional footballer who died of a heart attack during a recreational game on 21 June 2013 at the age of 23.