This thesis describes new approaches to extend the sensitivity reach of searches for three types of supersymmetric particles; staus, charginos and neutralinos, using the 139 fb⁻¹ of data collected with the ATLAS experiment at the Large Hadron Collider between 2015 and 2018. Searches for three different production modes of these particles are considered, with new approaches for each that highlight different methods for improving sensitivity.

Firstly, a search for direct stau production, in which staus decay to tau-leptons and neutralinos, is presented. The search makes use of Boosted Decision Trees to classify the signal from the Standard Model backgrounds and in doing so significantly improves the sensitivity compared to a previous search using the same dataset. No significant differences between the observed data and the Standard Model predictions are found but new, world-leading limits on direct stau production are set. Models with mass degenerate left and right-handed staus are excluded at 95% confidence level for stau masses between 80 GeV and 500 GeV, with the low mass reach closing a previous gap in sensitivity between previous searches performed by the ATLAS experiment and searches conducted by experiments at the Large Electron Positron collider. This search also provides the first sensitivity at the ATLAS experiment to the right-handed stau only interpretation, for which models with right-handed stau masses between 100 GeV and 350 GeV are excluded for a massless neutralino-one at 95% confidence level.

Secondly, new search channels for the production of charginos and neutralinos with highly compressed mass spectra are explored. In particular, a re-interpretation of a previous search for new physics performed at ATLAS in a final state with an energetic jet and large missing transverse momentum is compared to a sensitivity study of a vector boson fusion channel. The re-interpretation is found to have sensitivity to previously uncovered regions of parameter space; for a simplified wino/bino interpretation models with neutralino-two masses of 140 GeV are excluded at 95% confidence level, under the assumption that signal theoretical uncertainties have a negligible impact. The vector boson fusion channel is assessed to have comparatively limited sensitivity with differences in the signal modelling between this study and previous work on this channel being identified. However, it is shown that the use of machine learning in this channel can provide a significant gain in the sensitivity.

Thirdly, a statistical combination of previous ATLAS searches for chargino pair production in three different topologies is performed for the first time, leading to a notable enhancement in the overall sensitivity.