Let $X$ be a projective scheme over a finitely generated $k$-algebra $R$ such that $X$ is a smooth $n$-dimensional variety over $k$. We assume that $H$ is an ample and basepoint-free divisor, $D$ is a divisor with $h^{0}(X,\mathcal{O}(mD))\neq 0$ for some $m\in\mathbb{N}_{>0}$ and $E$ is a divisor such that the $\mathbb{Q}$-divisor $D-\lambda E$ is nef for some $\lambda\in\mathbb{Q}_{\geq 0}$. Then the sheaf $\mathcal{O}_{X}(K_{X/k}+E+dH)\otimes_{\mathcal{O}_{X}}\tau(\lambda\cdot\|D\|)$ is globally generated for all $d\geq n+1$.