A search for R-parity-conserving supersymmetry in events with large missing transverse momentum, jets and at least one hadronically decaying $τ$-lepton is presented. Both gluino and squark pair production are considered, with the cascade decay of each gluino or squark producing either a $τ$-slepton or a $τ$-sneutrino. Three channels are examined, requiring either exactly one hadronically decaying $τ$-lepton and no other leptons, exactly one hadronically decaying $τ$-lepton and at least one other lepton, or two or more hadronically decaying $τ$-leptons. Analyses in the three channels are optimised independently and combined statistically. Two separate analysis strategies, either a cut-and-count or machine-learning approach, are used. The search uses 140 $\mathrm{fb}^{-1}$ and 51.8 $\mathrm{fb}^{-1}$ of $pp$ collision data recorded by the ATLAS detector at the Large Hadron Collider during 2015-2018 at $\sqrt{s} = 13$ TeV and 2022-2023 at $\sqrt{s} = 13.6$ TeV, respectively. Gluino masses below $2.25$ TeV and squark masses up to $1.7$ TeV are excluded.