Factorization type probabilities of polynomials with prescribed coefficients over a finite field

Let $f(T)$ be a monic polynomial of degree $d$ with coefficients in a finite field $\mathbb{F}_q$. Extending earlier results in the literature, but now allowing $(q,2d)>1$, we give a criterion for $f$ to satisfy the following property: for all but $d^2-d-1$ values of $s$ in $\mathbb{F}_q$, the probability that $f(T)+sT+b$ is irreducible over $\mathbb{F}_q$ (as $b\in\mathbb{F}_q$ is chosen uniformly at random) is $1/d+O(q^{-1/2})$.