In 1992, Wagner proved that the Hadamard product of two totally positive lower triangular Toeplitz matrices is totally positive. In this work, we strengthen this result by establishing total monomial positivity for the Hadamard product of Jacobi--Trudi matrices. In particular, we resolve a conjecture of Sokal concerning the Hadamard square of Jacobi--Trudi matrices. Moreover, we provide a manifestly positive Schur expansion for the Hadamard square of Jacobi--Trudi matrices indexed by ribbons. In addition, we construct a corresponding representation, offering a representation-theoretic proof of the Schur positivity.