  Supersymmetric extensions of the Standard Model generally give a theoretical
upper limit on the lightest Higgs boson mass which may be uncomfortably close
to the current experimental lower bound of $m_h$ > 114 GeV. Here we show ways
in which this upper limit on $m_h$ can be relaxed substantially in the
perturbative regime, while maintaining the successful unification of gauge
couplings as observed in the minimal scenario. A class of models is presented,
which includes new vector-like matter with the quantum numbers of the 5, 5-bar
and singlet representations of SU(5) and having Yukawa couplings (\kappa) to
the usual Higgs doublet $H_u$. This new matter transforms non-trivially under a
"lateral" SU(N) gauge symmetry, which enables the new Yukawa couplings \kappa
to be sizable, even larger than the top-quark Yukawa coupling, and still be
perturbative at all momenta up to the unification scale of 10^{16} GeV. We find
that, consistent with low energy constraints, $m_h$ can be raised to values as
large as 200 - 300 GeV.
