 Restriction ampleness implies semiampleness for an effective divisor

If $E$ is an effective Cartier divisor on a projective scheme and ${\mathcal{O}}_E(E)$ is ample, then $E$ is semiample. The divisor restriction exact sequence and Serre vanishing make $H^1(X,(m-1)E)\to H^1(X,mE)$ surjective for large $m$. Their finite dimensions stabilize, so restriction on global sections is eventually surjective. Lift generators on $E$; off $E$, the canonical section of $mE$ generates. Together these generate ${\mathcal{O}}_X(mE)$, including on nonreduced $X$.