  Upper bounds on the three neutral scalar Higgs masses are considered in the
supersymmetric standard model with a gauge singlet Higgs field. When the
lightest Higgs is singlet-dominated the second lightest Higgs is shown to lie
near or below the theoretical upper bound on the lightest Higgs mass. We also
consider detectability of these Higgs bosons at a future $e^+ e^-$ linear
collider with $\sqrt{s}\sim 300 $ GeV and show that at least one of the neutral
scalar Higgs has a production cross section larger than 0.04~pb.
