Many of the best paying accounts are high st current accounts, though it does mean putting a bit in each
Open a Santander 123 Current Account and put £20,000 in that to get 1.5%. You get even more if you set up all your direct debits to go from that account.
I get around £30 or £35 a month interest on my account.
https://www.santander.co.uk/uk/current-accounts/123-current-account
Open a Nationwide FlexDirect current account and get 5% on the first £2,500 (as long as you pay in £1,000 each month)
https://www.nationwide.co.uk/products/current-accounts/flexdirect/features-and-benefits
Open a NatWest Savings Builder account and put £10,000 in that and get 1.5% (you have to put £50 in each month)
https://personal.natwest.com/personal/_compare_savings_accounts/_savings_builder.html
Open a Santander Regular ESaver account put £200 a month for a year and get 3% on that.
https://www.santander.co.uk/uk/savings/regular-esaver
Open a Nationwide Flex Regular Online Saver and put in £250 a month for a year and get 5% on that
https://www.nationwide.co.uk/products/savings/flex-regular-online-saver/features-and-benefits
I know this is a bit of hassle but they beat many of the normal "instant release" savings accounts that pay less than 1%