  Context. The companions of the exploding carbon-oxygen white dwarfs (CO WDs)
for producing type Ia supernovae (SNe Ia) are still not conclusively confirmed.
A red-giant (RG) star has been suggested to be the mass donor of the exploding
WD, named as the symbiotic channel. However, previous studies on the this
channel gave a relatively low rate of SNe Ia. Aims. We aim to systematically
investigate the parameter space, Galactic rates and delay time distributions of
SNe Ia from the symbiotic channel by employing a revised mass-transfer
prescription. Methods. We adopted an integrated mass-transfer prescription to
calculate the mass-transfer process from a RG star onto the WD. In this
prescription, the mass-transfer rate varies with the local material states.
Results. We evolved a large number of WD+RG systems, and found that the
parameter space of WD+RG systems for producing SNe Ia is significantly
enlarged. This channel could produce SNe Ia with intermediate and old ages,
contributing to at most 5% of all SNe Ia in the Galaxy. Our model increases the
SN Ia rate from this channel by a factor of 5. We suggest that the symbiotic
systems RS Oph and T CrB are strong candidates for the progenitors of SNe Ia.
