  I present approximate results that include third-order soft-gluon corrections
for the associated production of a single top quark with a $W$ boson. The
calculation uses expressions from soft-gluon resummation at
next-to-next-to-leading-logarithm (NNLL) accuracy. From the NNLL resummed cross
section I derive approximate next-to-next-to-next-to-leading order (aN$^3$LO)
cross sections for the process $bg \rightarrow tW^-$ at LHC and Tevatron
energies. The aN$^3$LO top-quark transverse-momentum and rapidity distributions
in $tW$ production are also presented.
