The discovery of 25 neutrinos coming from the SN1987A core-collapse supernova (CCSN) by the Super-Kamiokande, IMB and Baksan experiments marked the beginning of neutrino astronomy. A new observation of supernova neutrinos with current or upcoming experiments could provide key insight into the underlying mechanism of CCSNe, which is currently poorly understood. Due to the low interaction rate of neutrinos, experiments are however only sensitive to close-by supernovae. Since these events are quite rare, it is crucial to optimize the detection channels of all available experiments. In this contribution, a study of the backgrounds for CCSN searches at the KM3NeT neutrino telescope, currently under construction and taking data in the Mediterranean Sea, is presented. Using new dedicated observables, signatures from radioactive decays and atmospheric muons are modelled, and these results are compared to data to assess the quality of the modelling of the detector response to supernova neutrinos. Finally, based on these results, an improvement of $23\%$ in KM3NeT's distance horizon to a CCSN is presented.