  In this paper, we study the $a$-contraction property of small extremal shocks
for 1-d systems of hyperbolic conservation laws endowed with a single convex
entropy, when subjected to large perturbations. We show that the weight
coefficient $a$ can be chosen with amplitude proportional to the size of the
shock. The main result of this paper is a key building block in the companion
paper, [{arXiv:2010.04761}, 2020], in which uniqueness and BV-weak stability
results for $2\times 2$ systems of hyperbolic conservation laws are proved.
