Although type Ia supernovae (SNe Ia) are very useful in many astrophysical fields, their exact progenitor nature is still unclear. A basic method to distinguish the different progenitor models is to search the signal from the single-degenerate (SD) model, e.g., the signal for the existence of a nondegenerate companion before or after supernova explosion. Observationally, some SNe Ia show such signals, while the others do not. Here, we propose a universal model to explain these observations based on the spin-up/spin-down model, in which a white dwarf (WD) will experience a spin-down phase before supernova explosion, and the spin-down timescale is determined by its initial mass, i.e., the more massive the initial WD, the shorter the spin-down timescale and then the more likely the SN Ia is to show the SD signature. Therefore, our model predicts that the SNe Ia from hybrid carbon-oxygen-neon WDs are more likely to show the SD signature observationally, as some peculiar SNe Ia showed.