Squarefree polynomials with prescribed coefficients

For nonempty subsets $S_0, \dots, S_{n-1}$ of a (large enough) finite field $\mathbb{F}$ satisfying $$|S_1|, \dots, |S_{n-1}| > 2 \quad \mathrm{or} \quad |S_1|,|S_{n-1}| > n - 1,$$ we show that there exist $a_0 \in S_0, \dots, a_{n-1} \in S_{n-1}$ such that $$T^n + a_{n-1}T^{n-1} + \dots + a_1T + a_0 \in \mathbb{F}[T]$$ is a squarefree polynomial.