We propose new ways of robustifying goodness-of-fit tests for structural equation modeling under non-normality. These test statistics have limit distributions characterized by eigenvalues whose estimates are highly unstable and biased in known directions. To take this into account, we design model-based trend predictions to approximate the population eigenvalues. We evaluate the new procedures in a large-scale simulation study with three confirmatory factor models of varying size (10, 20, or 40 manifest variables) and six non-normal data conditions. The eigenvalues in each simulated dataset are available in a database. Some of the new procedures markedly outperform presently available methods. We demonstrate how the new tests are calculated with a new R package and provide practical recommendations.