  We study the stability of Schwarzschild black hole in Einstein-Weyl-scalar
(EWS) theory with a quadratic scalar coupling to the Weyl term. Its linearized
theory admits the Lichnerowicz equation for Ricci tensor as well as scalar
equation. The linearized Ricci-tensor carries with a regular mass term
($m^2_2$), whereas the linearized scalar has a tachyonic mass term
($-1/m^2_2$). It turns out that the double instability of Schwarzschild black
hole in EWS theory is given by Gregory-Laflamme and tachyonic instabilities. In
the small mass regime of $m_2<0.876$, the Schwarzschild black hole becomes
unstable against Ricci-tensor perturbations, while tachyonic instability is
achieved for $m_2<1.174$. The former would provide a single branch of
scalarized black holes, whereas the latter would induce infinite branches of
scalarized black holes.
