Version 1.6.0 of the code SusHi is presented. Concerning inclusive CP-even Higgs production in gluon fusion, the following new features with respect to previous versions have been implemented: expansion of the partonic cross section in the soft limit, i.e. around ; NLO QCD corrections in terms of the soft expansion; top-quark mass suppressed terms through NNLO; matching to the cross section at through NLO. For CP-even and -odd scalars, an efficient evaluation of the renormalization-scale dependence is included, and effects of dimension-5 operators can be studied, which we demonstrate for the SM Higgs boson and for a CP-even scalar with a mass of GeV. In addition, as a generalization of the previously available cross section, SusHi 1.6.0 provides the cross section for charged and neutral Higgs production in the annihilation of arbitrary heavy quarks. At fixed order in perturbation theory, SusHi thus allows to obtain Higgs cross-section predictions in different models to the highest precision known today. For the SM Higgs boson of GeV, SusHi yields pb for the gluon-fusion cross section at the LHC at TeV. Simultaneously, SusHi provides the renormalization-scale uncertainty of pb.