Three searches for the direct production of staus or charginos and neutralinos in final states with at least two hadronically decaying $\tau$-leptons are presented. For chargino and neutralino production, decays via intermediate staus or Wh bosons are considered. The analysis uses a dataset of $pp$ collisions corresponding to an integrated luminosity of 139\,fb$^{-1}$, recorded with the ATLAS detector at the Large Hadron Collider at a centre-of-mass energy of $13$ TeV. No significant deviation from the expected Standard Model background is observed and SUSY particle mass limits at 95\% confidence level are obtained in simplified models. For direct production of $\tilde{\chi}_1^+\tilde{\chi}_1^-$, chargino masses are excluded up to 970\,GeV, while $\tilde{\chi}_1^{\pm}$ and $\tilde{\chi}_2^0$ masses up to 1160\,GeV (330\,GeV) are excluded for $\tilde{\chi}_1^{\pm}\tilde{\chi}_2^0$/$\tilde{\chi}_1^+\tilde{\chi}_1^-$ production decaying via staus ($Wh$ bosons). Stau masses up to 480\,GeV are excluded for mass degenerate $\tilde{\tau}_{L,R}$ scenarios and up to 410\,GeV for $\tilde{\tau}_L$-only scenarios. The first sensitivity to $\tilde{\tau}_R$-only scenarios is presented here, with $\tilde{\tau}_R$ masses excluded up to 330\,GeV.