There is no rush in the sense of not having a mortgage, if you haven’t done anything by the time your fix ends you will still have a mortgage with your current provider - you will just move to their SVR (Standard Variable Rate). This is usually higher than the BoE base rate - and each mortgage provider sets their own SVR.
If you are looking at new rates now, and you may see some that are a slightly smaller percentage than your current provider. You need to bear in mind the costs and “faff” of changing provider. Yes, you may see a rate that is, say, 0.4% cheaper than your current provider. But just bear in mind that a new lender will, most likely, want to carry out a valuation, require proof of funds/income etc, Sometimes, a small percentage saving isn’t worth it - it could be easier and quicker with your current provider.
Either way, ensure you sort out a new deal - with current or new provider. Don’t just let things lapse and end up on the SVR - this is always more expensive.
Next time, please bear in mind that most lenders let you sort a new deal out a fair way before your current one ends - ie anything from 3-6 months in advance. Don’t leave it until you only have six weeks. And if, for example, you “choose” a deal 3 months before the end, if a better rate comes up before your new deal starts, you can usually reapply.