  The coexistence of various low-lying deformed states in $^{42}$Ca and
$\alpha$--$^{38}$Ar correlations in those deformed states have been
investigated using deformed-basis antisymmetrized molecular dynamics. Wave
functions of the low-lying states are obtained via parity and angular momentum
projections and the generator coordinate method (GCM). Basis wave functions of
the GCM calculation are obtained via energy variations with constraints on the
quadrupole deformation parameter $\beta$ and the distance between $\alpha$ and
$^{38}$Ar clusters. The rotational band built on the $J^\pi = 0_2^+$ (1.84 MeV)
state as well as the $J^\pi = 0_3^+$ (3.30 MeV) state are both reproduced. The
coexistence of two additional $K^\pi = 0^+$ rotational bands is predicted; one
band is shown to be built on the $J^\pi = 0_3^+$ state. Members of the
ground-state band and the rotational band built on the $J^\pi = 0_3^+$ state
contain $\alpha$--$^{38}$Ar cluster structure components.
