  Let $u\not\equiv -\infty$ be a subharmonic function on the complex plane
$\mathbb C$. In 2016, we obtained a result on the existence of an entire
function $f\neq 0$ satisfying the estimate $\log|f|\leq {\sf B}_u$ on $\mathbb
C$, where functions ${\sf B}_u$ are integral averages of $u$ for rapidly
shrinking disks as it approaches infinity. We give another equivalent version
of this result with $\log |f|\leq u$ outside a very small exceptional set if
$u$ is of finite order.
