  \alpha-(BEDT-TTF)2KHg(SCN)4 is considered to be in the charge-density-wave
(CDW) state below 8 K. We present new magnetoresistance data suggesting that
the material undergoes a series of field-induced CDW (FICDW) transitions at
pressures slightly exceeding the critical pressure Pc at which the zero-field
CDW state is destroyed. Further, we argue that a novel kind of FICDW
transitions, entirely determined by a superposition of the strong Pauli and
quantizing orbital effects of magnetic field on the CDW wavevector, arises when
the field is strongly tilted towards the conducting layers. These new
transitions can take place even in the case of a relatively well nested Fermi
surface. Finally we report on the superconducting (SC) state and its
coexistence with the CDW in the title compound under quasi-hydrostatic
pressure. Below Pc the material is most likely a heterogeneous SC/CDW mixture,
with the SC phase persisting down to ambient pressure. The SC onset temperature
appears to drastically increase upon entering the SC/CDW coexistence region.
