  Stability of Schwarzschild-AdS (SAdS) black hole is investigated in
Einstein-Weyl-scalar (EWS) theory with a negative cosmological constant. Here,
we introduce a quadratic scalar coupling to the Weyl term, instead of the
Gauss-Bonnet term. The linearized EWS theory admits the Lichnerowicz equation
for Einstein tensor as well as scalar equation. The linearized Einstein-tensor
carries with a regular mass term (${\cal M}^2$), whereas the linearized scalar
has a tachyonic mass term ($-3r_0^2/m^2r^6$). Two instabilities of SAdS black
hole in EWS theory are found as Gregory-Laflamme and tachyonic instabilities.
It shows that the correlated stability conjecture holds for small SAdS black
holes obtained from EWS theory by establishing a close relation between
Gregory-Laflamme and thermodynamic instabilities. On the other hand, tachyonic
instability of SAdS black hole can be used for making five branches of
scalarized black holes when considering proper thermodynamic quantities of EWS
theory (${\cal M}^2>0$).
