  The lighter neutral scalar Higgs mass is examined in gauge-mediated
supersymmetry-breaking models in which the messenger sector responsible for
SUSY breaking is allowed to involve more general sets of $SU(3) \otimes SU(2)
\otimes U(1)$ multiplets than those contained in the SU(5) $5 + \bar{5}$ or $10
+ \bar{10}$ multiplets. The largest mass for the lighter scalar Higgs is found
to be 132 GeV when the breaking parameter $\Lambda$ is taken to be 100<EMAIL_ADDRESS>Thus the predictions for the lightest Higgs mass can be tested at the upgraded
Tevatron runs for these general classes of GMSB models.
