In the $E_6$ inspired extension of the minimal supersymmetric (SUSY) standard model (MSSM) with an extra $U(1)_{N}$ gauge symmetry under
which right-handed neutrinos have zero charge, a single discrete $\tilde{Z}^{H}_2$ symmetry permits suppressing rapid proton decay and non-diagonal flavour transitions. To ensure anomaly cancellation this SUSY model (SE$_6$SSM) must include additional exotic matter beyond the MSSM.
If matter parity and $\tilde{Z}^{H}_2$ symmetry are preserved the SE$_6$SSM may involve two dark matter candidates. Here we consider the modification of the SE$_6$SSM in which the cold dark matter is composed of the lightest neutral exotic fermion and gravitino with mass $m_{3/2}\lesssim 1\,\mbox{GeV}$. When the sparticle mass scale lies beyond the multi-TeV range the phenomenologically acceptable density of the dark matter can be obtained in this case only for relatively low reheating temperatures $T_R\lesssim 10^{6-7}\,\mbox{GeV}$. Within this scenario the appropriate amount of the baryon asymmetry can be induced via the decays of the lightest right-handed neutrino/sneutrino into exotic states.
In general, the spin-independent part of the dark matter-nucleon scattering cross section can be much larger in the model under consideration than the corresponding experimental limit. Nevertheless we argue that there is a part of the SE$_6$SSM parameter space, in which this cross section is sufficiently strongly suppressed.