  Over a perfect field, we determine the sheaf of $\mathbb{A}^1$-connected
components of a class of threefolds given by the Blow-up of a variety admitting
a $\mathbb{P}^1$-fibration over either an $\mathbb{A}^1$-rigid or a
non-uniruled surface, along a smooth curve. As a consequence, we verify that
the sheaf of $\mathbb{A}^1$-connected components for such varieties is
$\mathbb{A}^1$-invariant.
