Abstract: We give algorithms for sampling several structured logconcave families to high accuracy. We further develop a reduction framework, inspired by proximal point methods in convex optimization, which bootstraps samplers for regularized densities to generically improve dependences on problem conditioning $\kappa$ from polynomial to linear. A key ingredient in our framework is the notion of a ``restricted Gaussian oracle'' (RGO) for $g: \mathbb{R}^d \rightarrow \mathbb{R}$, which is a sampler for distributions whose negative log-likelihood sums a quadratic (in a multiple of the identity) and $g$. By combining our reduction framework with our new samplers, we obtain the following bounds for sampling structured distributions to total variation distance $\epsilon$. \begin{itemize} \item For composite densities $\exp(-f(x) - g(x))$, where $f$ has condition number $\kappa$ and convex (but possibly non-smooth) $g$ admits an RGO, we obtain a mixing time of $O(\kappa d \log^3\tfrac{\kappa d}{\epsilon})$, matching the state-of-the-art non-composite bound \cite{LeeST20}. No composite samplers with better mixing than general-purpose logconcave samplers were previously known. \item For logconcave finite sums $\exp(-F(x))$, where $F(x) = \tfrac{1}{n}\sum_{i \in [n]} f_i(x)$ has condition number $\kappa$, we give a sampler querying $\widetilde{O}(n + \kappa\max(d, \sqrt{nd}))$ gradient oracles to $\{f_i\}_{i \in [n]}$. No high-accuracy samplers with nontrivial gradient query complexity were previously known. \item For densities with condition number $\kappa$, we give an algorithm obtaining mixing time $O(\kappa d \log^2\tfrac{\kappa d}{\epsilon})$, improving \cite{LeeST20} by a logarithmic factor with a significantly simpler analysis. We also show a zeroth-order algorithm attains the same query complexity. \end{itemize}