The anisotropy of the magnetoresistance has been measured systematically by changing the current and field direction with respect to the crystallographic axes in the quasi-two-dimensional organic conductor α-(BEDT-TTF)2KHg(SCN)4, where BEDT-TTF denotes bis(ethylenedithio)tetrathiafulvalene, which shows an antiferromagneticlike ordering as a spin-density wave below TA=10 K. The open direction of the Fermi surface changes from θ=0°in the normal-metal phase to 30°in the antiferromagneticlike phase (θ is the tilt angle from the c axis in the conductive a-c plane). The open direction in the normal-metal phase is in agreement with a tight-binding band-structure calculation based on the crystal structure at room temperature. Angle-dependent magnetoresistance oscillations (ADMRO's) have been measured in both phases using a double-axis sample rotation system in fields up to 26 T. In the normal-metal phase, the angle-dependent magnetoresistance oscillations show two-dimensional behavior resulting from the closed Fermi surface with cross-sectional area 14% of the Brillouin zone. On the other hand, in the antiferromagneticlike phase, the angle-dependent magnetoresistance oscillations show one-dimensional behavior, the open direction of which is along θ=30°as observed in the current-direction dependence. At 26 T higher than HA (T=1.5 K)=23 T, the phase boundary between the antiferromagneticlike phase and the normal-metal phase, we have observed exactly the same two-dimensional ADMRO effect as in the normal-metal phase at higher temperatures. In order to explain these anomalous temperature and field dependences of the Fermi surface, we propose a superlattice model which reconstructs the Fermi surface at the phase transition. This model shows that a pair of open Fermi surfaces in the normal-metal phase is nested to each other by the vector Q(=1/6ka+1/2kb+ 1) / 3 kc) and the new open Fermi surface is reconstructed in antiferromagneticlike phase by the multiconnected closed Fermi surface.