Let $D \in S$ be an ample class.
If $D^2 =6$ and there does not exist a class $F\in S$ such that
$F^2 = 0$ and $F\cdot D \in \{1,2\}$, then $$\varphi_D : X \longrightarrow \mathbb{P}^4$$ embeds $X$ as a degree $6$ surface.
If $D^2 = 8$ and there does not exist a class $F\in S$ such that
$F^2 = 0$ and $F\cdot D \in \{1,2,3\}$ then $$\varphi_D : X \longrightarrow \mathbb{P}^5 $$ either embeds $X$ as a generically transverse intersection of three quadrics with only rational double points, or realizes $X$ as a double cover of a Veronese surface.