I have just had a similar exercise, with another insurance company, not Tesco. Received my 2nd year renewal, checked on-line with same company, ticked the box which asked me if I already had insurance with them. I ticked the box stating that I was insured by them. Received a quotation, well below the renewal letter quotation.
Thinking this over, I decided to check with the insurances customer services. The adviser did a price check, and stated that she was unable to match the on-line price, even though it was the same company. After further discussion with the adviser, I decided to try elsewhere, and was able to obtain similar insurance cover at a much lower price through Argos.