Determinant of Hadamard product / sum of matrices (one diagonal)

Question

I am trying to compute the determinant of $\boldsymbol{W}\odot \boldsymbol{S}$, where $\boldsymbol{S} \in PD(p)$ positive semidefinite matrix and $\boldsymbol{W}$ is a matrix whose diagonal entries $w_ {jj}\geq 1$, and off diagonal entries $w_ {ij}=1$, $\forall i\neq j$; $i,j=1,\ldots,p$. Alternatively, the problem can be recast as follows: compute the determinant of $\boldsymbol{A} + \boldsymbol{S}$, where $\boldsymbol{A}=diag(a_{1},\ldots,a_{p})$ with $a_j=s_{jj}(w_{jj}-1)$. Is there a way the sought determinant can be expressed as a function of the quantities involved? Thanks.