  This paper is concerned with the elastic near cloaking for the Lam\'e system
in three-dimensions using the notion of elastic scattering coefficients (ESC).
Accordingly, the ESC of arbitrary three-dimensional objects are designed and
some of their properties are discussed using elements of the elastic layer
potential theory. Then, near-cloaking structures, coined as
ESC-vanishing-structures, are constructed for the elastic cloaking at a fixed
frequency or a band of frequencies. These multi-layered structures are designed
so that their first few significant ESC vanish ahead of
transformation-elastodynamics. The invisibility effect is achieved as the
arbitrary elastic object inside the cloak has a near-zero scattering
cross-section for a band of frequencies. The cloaking effect for the Lam\'e
system is significantly enhanced by the proposed near-cloaking structures.
