Stability of systems with a single delay and delay-dependent coefficients is studied along the line of the τ-decomposition approach. Criteria for determining crossing directions of imaginary characteristic roots with possibly mul-tiplicity are presented, with which system stability for any given delay value can be determined in a systematic way. In contrast to the previous research on this type of systems, our analysis is based on a novel two-parameter framework. With the new geometric insight, stronger criteria concerning the crossing direction of imaginary characteristic roots with possibly multi-plicity can be obtained using simplified and intuitive arguments. The stability analysis procedure is illustrated with an example inspired by biological applications.