Black-box variational inference (BBVI) scales poorly to high-dimensional problems when it is used to estimate a multivariate Gaussianapproximation with a full covariance matrix. In this paper, we extend the batch-and-match (BaM) framework for score-based BBVI toproblems where it is prohibitively expensive to store such covariance matrices, let alone to estimate them. Unlike classical algorithms forBBVI, which use stochastic gradient descent to minimize the reverse Kullback-Leibler divergence, BaM uses more specialized updatesto match the scores of the target density and its Gaussian approximation. We extend the updates for BaM by integrating them with a more compact parameterization of full covariance matrices. In particular, borrowing ideas from factor analysis, we add an extra step toeach iteration of BaM---a patch---that projects each newly updated covariance matrix into a more efficiently parameterized family of diagonal plus low rank matrices. We evaluate this approach on a variety of synthetic target distributions and real-world problems inhigh-dimensional inference.