  We introduce an improvement to the FxFx matrix element merging procedure for
$pp\to t \bar t W$ production at NLO in QCD with one and/or two additional
jets. The main modification is an improved treatment of jets that are not
logarithmically enhanced in the low transverse-momentum regime. We provide
predictions for the inclusive cross section and the $t \bar t W$ differential
distributions including parton-shower effects. Taking also the NLO EW
corrections into account, this results in the most-accurate predictions for
this process to date. We further proceed to include the on-shell LO decays of
the $t \bar t W$ including the tree-level spin correlations within the
narrow-width approximation, focusing on the multi-lepton signatures studied at
the LHC. We find a $\sim\!\!30\%$ increase over the NLO QCD prediction and
large non-flat $K$-factors to differential distributions.
