  We investigate an extension of the MSSM Higgs sector by including the effects
of all dimension-five and dimension-six effective operators and their
associated supersymmetry breaking terms. The corrections to the masses of the
neutral CP-even and CP-odd Higgs bosons due to the d=5 and d=6 operators are
computed. When the d=5 and d=6 operators are generated by the same physics
(i.e. when suppressed by powers of the same scale M), due to the relative
$\tan\beta$ enhancement of the latter, which compensates their extra scale
suppression (1/M), the mass corrections from d=6 operators can be comparable to
those of d=5 operators, even for conservative values of the scale M. We
identify the effective operators with the largest individual corrections to the
lightest Higgs mass and discuss whether at the microscopic level and in the
simplest cases, these operators are generated by "new physics" with a sign
consistent with an increase of m_h. Simple numerical estimates easily allow an
increase of m_h due to d=6 operators alone in the region of 10-30 GeV, while
for a much larger increase light new states beyond MSSM may be needed, in which
case the effective description is unreliable. Special attention is paid to the
treatment of the effective operators with higher derivatives. These can be
removed by non-linear field redefinitions or by an "unfolding" technique, which
effectively ensure that any ghost degrees of freedom (of mass $\geqsim M$) are
integrated out and absent in the effective theory at scales much smaller than
M. Considering general coefficients of the susy operators with a scale of new
physics above the LHC reach, it is possible to increase the tree-level
prediction for the Higgs mass to the LEPII bound, thus alleviating the MSSM
fine-tuning.
