$SCBS$ property (which means smallness up to complemented Banach subspace) enables us to prove that ``a bounded perturbation of an automorphism on a
complete barrelled locally convex space $X$ is stable up to some complemented Banach subspace $B$". We understand that it is not a necessary condition to have
this stability mentioned above.~ We characterize quasi-normable $l$-K\"othe spaces and give a sufficient condition to asymptotical normability in Fr\'echet spaces in
order to extend the range of the $SCBS$ property in Fr\'echet and $DF$ spaces. In fact we show that quasi-normable $l$-K\"othe spaces, locally convex direct
sums of Banach spaces and strong duals of some asymptotically normable Fr\'echet spaces have the $SCBS$ property.~ If we don't require complementedness of
the Banach subspace $B$ in the $SCBS$ property (that is the $SBS$ property) we are able to show that ``a quasi-barreled $DF$ space $X$ with its strong dual
satisfying the openness condition has the $SBS$ property", hence trivially every strict $LB$-space has the property $SBS$. Moreover, while investigating the
$SBS$ property we obtain a new characterization to the openness condition in both Fr\'echet and $DF$ spaces.

Keywords : {$SCBS$ Property, $SBS$ Property, Bounded Factrorization Property, Quasi-diagonal Operators, Quasi-normable Spaces,