We study a family of non-Hermitian tetradiagonal Toeplitz matrices having a limiting set consisting of one analytic arc only. We derive individual asymptotic expansions for all eigenvalues as the matrix size grows to infinity. Additionally, we provide specific expansions for the extreme eigenvalues, which are those approaching the endpoints of the limiting set. Although this family does not belong to the simple-loop class, we managed to extend the existing theory to this case. Our results reveal the intricate details of the eigenvalue structure and allow a high accuracy direct calculation.