Varieties with a reducible hyperplane section whose two components are hypersurfaces

We classify smooth complex projective varieties $X\subset\mathbb{P}^N$ of dimension $n\geq 2$ admitting a divisor of the form $A+B$ among their hyperplane sections, both $A$ and $B$ of codimension $\leq 1$ in their respective linear spans. In this setting, one of the following holds: 1) $X$ is either the Veronese surface in $\mathbb{P}^5$ or its general projection to $\mathbb{P}^4$, 2) $n\leq 3$ and $X\subset{\mathbb{P}}^{n+2}$ is contained in a quadric cone of rank $3$ or $4$, 3) $n=2$ and $X\subset\mathbb{P}^3$.