Cell-aware test (CAT) explicitly targets faults caused by cell-internal short and open defects and has been shown to significantly reduce test escape rates. CAT library cell characterization is typically done for only two defect resistance values: one representing hard opens and another one representing hard shorts. In this paper, similar to fishermen tightening the mesh size of their nets to catch small fish, we perform library characterization as efficiently as possible for a set of resistances representing increasingly weaker defects, and then adjust our ATPG flow to explicitly target faults caused by the weakest still-detectable variant of each potential defect. We implemented this novel approach in an experimental ATPG tool flow script, using functions of Cadence's Modus as building blocks. To assess the effectiveness of our approach, we formulate a new dedicated test metric: the weakest fault coverage wfc. Compared to conventional CAT targeting hard defects only, experimental results show that our new approach enhances detection of weakest faults and significantly reduces wfc escapes =1-wfc, while maintaining its original (hard-defect) fault coverage fc, of course at the expense of (acceptable) increases in the required number of test patterns and associated test generation time.