We report on our recent full off-shell $pp \rightarrow t\bar{t}W^+j+X$ calculation at the NLO level in QCD. Unstable resonant particles like the $W^\pm/Z$ vector bosons and top quarks are modelled with Breit-Wigner propagators. Furthermore double-, single- and non-resonant contributions as well as all their interference terms are consistently included at the matrix element level. We provide integrated and differential (fiducial) cross-section results and investigate the size of theoretical uncertainties as well as the impact of renormalization- and factorization scale settings on the theoretical predictions. Finally, we explore the effects of additional jet activity in $t\bar{t}W^+$ production and top-quark decays.