In a recent paper Melman [12] derived upper bounds for the smallest eigenvalue of a real symmetric Toeplitz matrix in terms of the smallest roots of rational and polynomial approximations of the secular equation $f(lambda)=0$, the best of which being constructed by the $(1,2)$-Pad{accent19 e} approximation of $f$. In this paper we prove that this bound is the smallest eigenvalue of the projection of the given eigenvalue problem onto a Krylov space of $T_n^{-1}$ of dimension 3. This interpretation of the bound suggests enhanced bounds of increasing accuracy. They can be substantially improved further by exploiting symmetry properties of the principal eigenvector of $T_n$.