  We consider the POWHEG generator for a H/W/Z boson plus one jet, augmented
with the recently proposed MiNLO method for the choice of scales and the
inclusion of Sudakov form factors. Within this framework, the generator covers
all the transverse-momentum region of the H/W/Z boson, i.e. no generation cuts
are needed to obtain a finite result. By construction, the generator achieves
NLO accuracy for distributions involving a finite (and relatively large)
transverse momentum of the boson. We examine the conditions under which also
the totally inclusive distributions (e.g. the boson rapidity distribution)
achieve NLO accuracy. We find that a minimal modification of the MiNLO
prescription is sufficient to achieve such accuracy. We thus construct a NLO
generator for H/W/Z boson plus one jet production such that it smoothly merges
into a NLO single boson production in the small transverse-momentum region. We
notice that, by simply reweighting the boson rapidity distribution to NNLO
predictions, we achieve a NNLO accurate generator matched to a shower. The
approach applies to all production processes involving a colorless massive
system plus one jet. We discuss how it may be extended to general processes.
