  This paper is concerned with investigating the asymptotic behavior of the
gradients of solutions to a class of elliptic systems with general boundary
data, especially covering the Lam\'{e} systems, in a narrow region. The novelty
of this paper lies in finding the correction term to improve the previous
gradient estimates to be a precise asymptotic formula. Through this
justification, we capture all singular terms and thus give a fairly sharp
characterization in terms of the singularities of the gradients.
