Number of polynomials with the same splitting field

Question

From the Chebotarev density theorem, we know that if we fix a $S_n$ extension $K/\mathbb{Q}$, a polynomial $f$ have the same splitting field if and only if it splits completely for every prime splitting completely in $K/\mathbb{Q}$ except finitely many. So for those primes we know the behaviour of $f\mod p$. My question is, if the Frobenius at a prime $p$ in $K/\mathbb{Q}$ has another splitting behaviour (maybe starting with the case of a transposition), does it constrain $f\mod p$? My ultimate goal would be to upper bound the probability that an integer polynomial of height:=max of |.| of the coefficients $\le S$ of the has splitting field $K$, as $S\rightarrow+\infty$.