I've had my Jazz since 2012 and have been breakdown-covered by the Honda supplied cover. I suspended my AA membership on buying the car, but it had less than a month to go on it.
Coming to the end of my Honda cover, I phoned to re-activate the AA cover and sort out some more for the coming year. Quoted £225, I pointed out that a new AA member would only have to pay £100 for the same cover. I asked how long I would need to let my cover lapse for before the AA would consider me a new customer and was told that they could possibly negotiate a lower price for me if I left because their cover was too expensive.
Now quoted £115, I said that the extra £15 would buy me a fair few miles of petrol, so after "consultation", they reduced the price to £100, matching the new member price.
Not boasting, but just to say it might be worth a nice calm phone call if you're renewing with the AA. My hat is doffed to them for demonstrating flexibility.