  The LUX experiment has recently set very strong constraints on
spin-independent interactions of WIMP with nuclei. These null results can be
accommodated in NMSSM provided that the effective spin-independent coupling of
the LSP to nucleons is suppressed. We investigate thermal relic abundance of
singlino-higgsino LSP in these so-called spin-independent blind spots and
derive current constraints and prospects for direct detection of spin-dependent
interactions of the LSP with nuclei providing strong constraints on parameter
space. We show that if the Higgs boson is the only light scalar the new LUX
constraints set a lower bound on the LSP mass of about 300 GeV except for a
small range around the half of $Z^0$ boson masses where resonant annihilation
via $Z^0$ exchange dominates. XENON1T will probe entire range of LSP masses
except for a tiny $Z^0$-resonant region that may be tested by the LZ
experiment. These conclusions apply to general singlet-doublet dark matter
annihilating dominantly to $t\bar{t}$. Presence of light singlet
(pseudo)scalars generically relaxes the constraints because new LSP (resonant
and non-resonant) annihilation channels become important. Even away from
resonant regions, the lower limit on the LSP mass from LUX is relaxed to about
250 GeV while XENON1T may not be sensitive to the LSP masses above about 400
GeV.
