  We show that the bilateral backward shift on $\ell^p(\mathbb{Z},\omega)$ that
has a projective orbit with a non-zero limit point is supercyclic. This
phenomenon holds also for $\Gamma$-supercyclicity, which extends a result
obtained for the first time by Chan and Seceleanu. Moreover, we show that if
$K$ is a compact subset of $\ell^p(\mathbb{N},\omega)$ such that its orbit
under the unilateral backward shift $B$ on $\ell^p(\mathbb{N},\omega)$ has a
non-zero weak limit point, then $B$ is hypercyclic.
