In this paper we consider a family of tetradiagonal (= four non-zero diagonals) Toeplitz matrices with a limiting set consisting in one analytic arc only and obtain individual asymptotic expansions for all the eigenvalues, as the matrix size goes to infinity. Additionally, we provide specific expansions for the extreme eigenvalues which are the eigenvalues approaching the extreme points of the limiting set. In contrast to previous related works, we study non-Hermitian Toeplitz matrices having non-canonical distribution and a real limiting set. The considered family does not belong to the so-called simple-loop class, nevertheless we manage to extend the theory to this case. The achieved formulas reveal the fine details of the eigenvalue structure and allow us to directly calculate high accuracy eigenvalues, even for matrices of relatively small size.