  We study the acylindrical hyperbolicity of the outer automorphism group of a
right-angled Artin group $A_\Gamma$. When the defining graph $\Gamma$ has no
SIL-pair (separating intersection of links), we obtain a necessary and
sufficient condition for $\mathrm{Out}(A_\Gamma)$ to be acylindrically
hyperbolic. As a corollary, if $\Gamma$ is a random connected graph satisfying
a certain probabilistic condition, then $\mathrm{Out}(A_\Gamma)$ is not
acylindrically hyperbolic with high probability. When $\Gamma$ has a maximal
SIL-pair system, we derive a classification theorem for partial conjugations.
Such a classification theorem allows us to show that the acylindrical
hyperbolicity of $\mathrm{Out}(A_\Gamma)$ is closely related to the existence
of a specific type of partial conjugations.
