We obtain an improved upper bound on the lightest neutral CP-even Higg
s boson mass from a low energy renormalisation-group analysis of the H
iggs sector of the next-to-minimal supersymmetric standard model. We f
ind m(h) < 145 GeV for m(t) = 90 GeV, decreasing to m(h) < 123 GeV for
m(t) = 180 GeV. We also discuss the light Higgs spectrum in the regio
n of parameter space close to the upper bound.