The construction of tangent bundle on a $C^\infty$ manifold, as for example in the book of Warner, uses the existence of double derivatives. Of course the tangent space for a point is first constructed in case of an open set in a Euclidean space and then the whole setup is glued up. But then in the neighborhood of a point the second derivatives might be different for a different $C^1$-chart around the point. So I suppose the tangent space/tangent bundle construction is for $C^2$-manifolds.
Since $C^2$ is apparently the minimum condition for existence of tangent bundle, is the tangent bundle just $C^1$, or it is again $C^2$?