  We report high magnetic field linear magnetostriction experiments on
CeCoIn$_5$ single crystals. Two features are remarkable: (i) a sharp
discontinuity in all the crystallographic axes associated with the upper
superconducting critical field $B_{c2}$ that becomes less pronounced as the
temperature increases; (ii) a distinctive second order-like feature observed
only along the c-axis in the high field (10 T $ \lesssim B \leq B_{c2}$) low
temperature ($T \lesssim$ 0.35 K) region. This second order transition is
observed only when the magnetic field lies within 20$^o$ of the ab-planes and
there is no signature of it above $B_{c2}$, which raises questions regarding
its interpretation as a field induced magnetically ordered phase. Good
agreement with previous results suggests that this anomaly is related to the
transition to the Fulde-Ferrel-Larkin-Ovchinnikov superconducting state.
