We determine the Schouten–Van Kampen connection associated to the Levi-Civita connection of a general natural metric on the total space TM of the tangent bundle of a Riemannian manifold. We provide the necessary and sufficient conditions for the obtained Schouten–Van Kampen connection to be torsion free and then to coincide with the Levi-Civita connection. We characterize the general natural α-structures on TM, which are parallel with respect to the torsion free Schouten-Van Kampen connection. Finally, we obtain the (para-)K¨ahler structures of general natural lift type on TM, for which the α-structure is parallel with respect to both Levi-Civita and Schouten-Van Kampen connection.