On 16 July 2015 Bambock moved abroad for the first time in his career, signing a two-year deal with Spanish Segunda División side SD Huesca.[3][4] He made his professional debut on 22 August, coming on as a second-half substitute for Iñigo Ros in a 2–3 home loss against Deportivo Alavés.[5]
On 12 July 2017, Bambock signed a one-year deal with Eredivisie side Sparta Rotterdam.[6] After appearing rarely and mainly for the B-side, he cut ties with the club the following February.[7]
On 1 February 2018, Bambock returned to Spain and its second level after signing for Córdoba CF,[8] but registration problems meant he could not feature for the side during the remainder of the season.[9] He was finally included in the B-team in Segunda División B in April,[10] being only registered to the first-team on 16 August.[11]
On 18 January 2019, Bambock terminated his contract with the Blanquiverdes,[12] and joined Israeli side Maccabi Petah Tikva FC four days later.
On 21 June 2019, Bambock signed a contract with Marítimo.[13]
On 12 August 2021, he joined Grenoble on a two-year contract.[14]
On 21 June 2023, it was announced that Bambock signed a contract with OTP Bank Liga club Fehérvár. He will play in jersey number 6, and signed a contract until 30 June 2025.[15] In January 2024, he joined Super League Greece club Panetolikos on loan for the remainder of the season.[16]