  We report $^{115}$In nuclear magnetic resonance (NMR) measurements in
CeCoIn$_5$ at low temperature ($T \approx 70$ mK) as a function of magnetic
field ($H_0$) from 2 T to 13.5 T applied perpendicular to the $\hat c$-axis.
NMR line shift reveals that below 10 T the spin susceptibility increases as
$\sqrt{H_0}$. We associate this with an increase of the density of states due
to the Zeeman and Doppler-shifted quasiparticles extended outside the vortex
cores in a d-wave superconductor. Above 10 T a new superconducting state is
stabilized, possibly the modulated phase predicted by Fulde, Ferrell, Larkin
and Ovchinnikov (FFLO). This phase is clearly identified by a strong and linear
increase of the NMR shift with the field, before a jump at the first order
transition to the normal state.
