Recent studies have acknowledged the many benefits of including fossils in phylogenetic inference (e.g., reducing long-branch attraction). However, unstable taxa are known to be problematic, as they can reduce either the resolution of the strict consensus or branch support. In this study, we evaluate whether unstable taxa that reduce consensus resolution affect support values, and the extent of such impact, under equal and extended implied weighting. Two sets of analyses were conducted across 30 morphological datasets to evaluate complementary aspects. The first focused on the analytical conditions incrementing the terminal instability, while the second assessed whether pruning wildcards improves support. Changes in support were compared with the “number of nodes collapsed by unstable terminals,” their “distance to the root,” the “proportion of missing data in a dataset,” and the “proportion of sampled characters.” Our results indicate that the proportion of missing entries distributed among closely related taxa (for a given character) might be as detrimental for stability as those distributed among characters (for a given terminal). Unstable terminals that (1) collapse few nodes or (2) are closely located to the root node have more influence on the estimated support values. Weighting characters according to their extra steps while assuming that missing entries contribute to their homoplasy reduced the instability of wildcards. Our results suggest that increasing character sampling and using extended implied weighting decreases the impact of wildcard terminals. This study provides insights for designing future research dealing with unstable terminals, a typical problem of paleontological data.