Quasi-Sasakian manifolds are an example of what you ask. They include Sasakian manifolds, that is normal contact metric manifolds, as well as normal almost contact manifolds with closed fundamental 2-forms. More generally, almost contact metric manifolds include contact and almost contact metric manifolds.

Please see Blair's book on contact manifold. There you will get answers. Perhaps Springer is the publisher.

Thank you for your suggestion Professors Nicola and Bhowmik

Professor Blair's book "Riemannian geometry of contact and symplectic manifolds" was published by Birkhauser, not Springer.

Very thanks for information to Nicola, bhowmik and Chen sir.