Background: I got caught out a few years ago by Nationwide increasing my home insurance premium year on year, way over the price offered to new customers. It was a wakeup call when I realised how much I'd been overpaying, so ever since then I have ended my policy each year and taken out a new one (still with Nationwide), usually saving around £30 on the premium. The Government are planning to ban the practice of overcharging existing customers (see here: www.moneysavingexpert.com/news/2020/09/insurers-face-ban-on-hiking-prices-for-existing-customers-at-ren/), but not until next year.
This year my insurance was due to end on Aug 20th so, as usual, I requested a new customer quote and, as usual, it was cheaper (by £40). But then I realised I should have started the new policy on the 21st instead. I updated just that one detail and suddenly the quote changed to £5 higher than my renewal quote. It looks like their algorithm has recently been altered to recognise existing customers looking for new-joiner price cut and is increasing the price on that basis to give the appearance that they're no longer overcharging existing customers. 
I tried to purchase the cheaper quote online, but got an error, probably because the policy overlapped my existing policy by one day - I had to phone up the call centre in order to do it. That was successful, and I was told that the overlap wouldn't be a problem.