  We study the geodesic X-ray transform $I_\Gamma$ of tensor fields on a
compact Riemannian manifold $M$ with non-necessarily convex boundary and with
possible conjugate points. We assume that $I_\Gamma$ is known for geodesics
belonging to an open set $\Gamma$ with endpoint on the boundary. We prove
generic s-injectivity and a stability estimate under some topological
assumptions and under the condition that for any $(x,\xi)\in T^*M$, there is a
geodesic without conjugate points in $\Gamma$ through $x$ normal to $\xi$.
