Plotting two-parameter confidence regions is non-trivial. Numerical methods often rely on a computationally expensive grid-like exploration of the parameter space. A recent advance reduces the two-dimensional problem to many one-dimensional problems employing a trigonometric transformation that assigns an angle from the maximum likelihood estimator, and an unknown radial distance to its confidence region boundary. This paradigm shift improves computation time by orders of magnitude, but it is not robust. Specifically, parameters differing greatly in magnitude and/or challenging non-convex confidence region shapes make the plot susceptible to inefficiencies and/or inaccuracies. This article improves the technique by (1) keeping confidence region boundary searches in the parameter space, (2) selectively targeting confidence region boundary points in lieu of uniformly-spaced angles from its maximum likelihood estimator, and (3) enabling access to regions otherwise unreachable due to multiple roots at select angles. Each algorithm is automated and publicly available via the R conf package, whose functionality and graphics are highlighted in this presentation.