Given a couple of smooth positive measures of same total mass on a compact connected Riemannian manifold $M$, we look for a smooth optimal transportation map $G$, pushing one measure to the other at a least total squared distance cost, directly by using the continuity method to produce a classical solution of the elliptic equation of Monge--Ampère type satisfied by the potential function $u$, such that $G =\exp(\grad u)$. This approach boils down to proving an \textit{a priori} upper bound on the Hessian of $u$, which was done on the flat torus by the first author. The recent local $C^2$ estimate of Ma--Trudinger--Wang enabled Loeper to treat the standard sphere case by overcoming two difficulties, namely: in collaboration with the first author, he kept the image $G(m)$ of a generic point $m\in M$, uniformly away from the cut-locus of $m$; he checked a fourth-order inequality satisfied by the squared distance cost function, proving the uniform positivity of the so-called $c$-curvature of $M$. In the present paper, we treat along the same lines the case of manifolds with curvature sufficiently close to 1 in $C^2$ norm -- specifying and proving a conjecture stated by Trudinger.