As the ongoing operations of the Large Hadron Collider continue to push the boundaries of high-energy particle physics, there arises a compelling need to revisit our search strategies for potential new physics phenomena at the TeV scale. A considerable amount of work has been dedicated to the development of novel reconstruction techniques, one of which is presented in this thesis; the formulation and investigation of an adaptive multi-vertex fitting and finding algorithm. In the process, this endeavour unveiled an opportunity for a new study. In this study, the focus centres on the investigation of final states characterised by two displaced leptons that neither originate from the primary interaction point nor are connected to a common vertex. This signature provides sensitivity to new theoretical particles, sleptons, arising from an extension of the Standard Model called Supersymmetry (SUSY). The region of interest is an unexplored gap in phase space for LHC-accessible SUSY, existing between final states with leptons with large displacement and prompt leptons. From 139 fb−1 of √s = 13 TeV proton-proton collision data collected by the ATLAS detector, events with oppositely charged lepton pairs with millimetre-range impact parameters are selected. The background contributions are estimated using a data- driven background estimation method. Expected sensitivity to sleptons in final states with electrons is presented and the unblinded results in final states involving two muons. Results are consistent with the background hypothesis, therefore limits on slepton lifetimes and masses are set in the di-muon channel.