The progenitors of type Ia supernovae (SNe Ia) are still under debate. There are two popular progenitor models for the formation of SNe Ia, i.e. the double-degenerate (DD) model and the single-degenerate (SD) model. However, none of these models can explain all of the observed properties of SNe Ia. In this paper, we introduce the white dwarf+helium (WD+He) subgiant channel in the context of the DD model and the semidetached WD+red-giant (RG) channel in the context of the SD model: (1) after considering the WD+He subgiant channel, the theoretical delay-time distribution of SNe Ia from the DD model can reproduce the observed results better;
(2) by using an updated mass-transfer method for the semidetached WD+RG systems (varying with the local material states), the initial parameter space and SN Ia birthrate from the semidetached WD+RG channel is enlarged significantly.