Was also going to suggest the Cinnamon Trust also.I dog walk for a living but when I scale back my business will look into some voluntary walking and think this is a brilliant idea to help people who otherwise would have the heartbreaking decision of re-homing much loved pets.
Definitely volunteer for RSPCA - I've done this and loved it! I wouldn't just advertise it for free - why not offer it as a paid service (at a low rate if you dn't need the money) and then donate the money to a dog rehoming charity.
The Cinnamon Trust are fab - they enabled my great aunt to keep her beloved dog to the end of her life.
Many smaller rescues appreciate any time a volunteer can give them to walk and socialise the dogs - the greyhound rescue we looked at were very happy for anyone to rock up and take some hounds for a walk on their land without any commitment.