Context. The companions of the exploding carbon-oxygen white dwarfs (CO WDs) that produce type-Ia supernovae (SNe Ia) have still not been conclusively identified. A red-giant (RG) star can fill this role as the mass donor of the exploding WD - this channel for producing SNe Ia has been named the symbiotic channel. However, previous studies on this channel have given 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 arising from the symbiotic channel under 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. First, we obtain the parameter space that leads to SNe Ia by evolving a large number of semidetached WD + RG systems with the Eggleton stellar-evolution code. Second, we investigate the Galactic rates and delay-time distributions of SNe Ia using a binary population synthesis method. Results. The parameter space of WD+RG systems that can produce SNe Ia is enlarged significantly judging by our calculations. This channel could produce SNe Ia with intermediate and old ages, contributing up to 5% of all SNe Ia in the Galaxy. Our model increases the SN Ia rate from this channel by a factor of five. We suggest that the symbiotic systems RS Oph and T CrB are strong candidates for the progenitors of SNe Ia.