  We investigate the tachyonic instability of Kerr-Newman (KN) black hole with
a rotation parameter $a$ in the Einstein-Chern-Simons-scalar theory coupled
with a quadratic massive scalar field. This instability analysis corresponds to
exploring the onset of spontaneous scalarization for KN black holes. First, we
find no $a$-bound for $\alpha<0$ case by considering (1+1)-dimensional
analytical method. A direct numerical method is adopted to explore
(2+1)-dimensional time evolution of a massive scalar perturbation with positive
and negative $\alpha$ to obtain threshold curves numerically. We obtain
threshold curves $\alpha_{\rm th}(a)$ of tachyonic instability for positive
$\alpha$ without any $a$-bounds. We expect to find the same threshold curves
$\alpha_{\rm th}(a)$ of tachyonic instability for negative $\alpha$ without any
$a$-bound because its linearized scalar theory is invariant under the
transformation of $\alpha\to -\alpha $ and $\theta\to -\theta$. In addition, it
is found that the scalar mass term suppresses tachyonic instability of KN black
holes.
