If $M$ is a smooth manifold, let $AM$ be the subspace of the double tangent bundle $TTM$ consisting of vectors $v$ such that $\pi^{TM}(v) = \pi^X_*(v)$, where $\pi^X: TX \to X$ is the projection of a tangent bundle. It is a bundle of affine spaces on $TM$, modeled on the "tautological" bundle $TM \times_M TM$. The significance of $AM$ is that if $\gamma$ is a smooth curve on $M$, its second derivative $\ddot \gamma$ lies in $AM$.
If the manifold $M$ is equipped with a section of $AM$ defined on all of $TM$, we can define a sort of covariant derivative of the tangent vector of a curve, by taking the second derivative and subtracting the fixed section. In particular, we get a notion of geodesic. So it seems that such a section plays some of the roles of a connection on $TM$.
Do these objects have a name? Where can I learn more about them? Can I think of them as connections in some more general sense?