Let $\Omega,\Lambda\subset\mathbb R^n$ be bounded convex domains, and let $\mu_0=\rho_0\,d\mathcal L^n$ and $\mu_1=\rho_1\,d\mathcal L^n$ satisfy uniform density bounds. Suppose that, on compact subsets of $\Omega$ and $\Lambda$, the densities $\rho_0$ and $\rho_1$ are $C^{k,\alpha}$ for some integer $k\ge 0$ and $0<\alpha<1$, with $\rho_1$ bounded away from zero. Assume also that the Brenier potential $\phi$ is an Alexandrov solution, is strictly convex on compact subsets of $\Omega$, and has sections compactly contained in $\Omega$ whose gradient images stay in compact subsets of $\Lambda$. Then, for every compact set $K\subset\Omega$, there is a compact set $K'\subset\Lambda$ with $\nabla\phi(K)\subset K'$ and the equation \begin{align*} \det D^2\phi(x)=\frac{\rho_0(x)}{\rho_1(\nabla\phi(x))} \end{align*} implies $\phi\in C^{k+2,\alpha}(K)$, with estimates depending on $K$, $K'$, the density bounds, the local $C^{k,\alpha}$ norms of $\rho_0$ and $\rho_1$, and the quantitative strict-convexity regime. In particular, under the corresponding smooth positive-data and quantitative interior convexity hypotheses, the transport map $\nabla\phi$ is smooth in the interior.