  We propose the definition of $\ell$-away ACM bundle on a polarized variety
$(X, \mathcal{O}_{X}(h))$. Then we give constructions of $\ell$-away ACM
bundles on $(\mathbb{P}^2 , \mathcal{O}_{\mathbb{P}^2}(1))$, $(\mathbb{P}^1
\times \mathbb{P}^1, \mathcal{O}_{\mathbb{P}^1 \times \mathbb{P}^1}(1,1))$ and
the anticanonically polarized blow up of $\mathbb{P}^2$ up to three non
collinear points. Also, we give the complete classification of $\ell$-away ACM
bundles $\mathcal{E}$ of rank 2 for values $1 \leq \ell \leq 2$ on
$(\mathbb{P}^2 , \mathcal{O}_{\mathbb{P}^2}(1))$. Similarly, on $(\mathbb{P}^1
\times \mathbb{P}^1, \mathcal{O}_{\mathbb{P}^1 \times \mathbb{P}^1}(1,1))$, we
give such a classification if $\mathrm{det}(\mathcal{E}) =
\mathcal{O}_{\mathbb{P}^1 \times \mathbb{P}^1}(a,a)$ for some $a \in
\mathbb{Z}$. Moreover, we prove that the corresponding graded module
$\mathrm{H}_*^1 ( \mathcal{E}) = \underset{{t \in \mathbb{Z} }}{\bigoplus}
\mathrm{H}^1 (\mathcal{E} (th))$ is connected, extending the similar result for
bundles on $\mathbb{P}^2$.
