You could look into getting an offset mortgage.
This means that you have a mortgage which has a savings account attached to it. You don't technically receive interest on the savings account. However, when you have money in the savings account, it reduces the amount of interest you pay on your mortgage.
For example, if you have a mortgage of £100,000 and pay 3% interest on it, then normally you pay £3000 interest per year. However, if it's an offset mortgage and you have £50,000 in the linked savings account, then the amount of mortgage which you pay interest on is reduced by that £50,000. So (in this example) the amount of interest you pay is halved, to £1500 per year. Obviously this is a very simplified example, but you can see how it works.
The advantages are similar to the advantages of overpaying your mortgage, except that you can get the money out of the offset savings account easily whenever you want. So if you suddenly need cash, it's available.
It also has advantages over having a separate interest-paying savings account, because you would probably have to pay tax on the interest you received from a separate savings account, whereas there is no tax to pay on the offset savings account (because it doesn't technically pay you any interest, it just reduces the interest you pay on your mortgage.) Plus most high interest savings accounts require you to leave your money in there for at least 2 years without touching it, so it's not so flexible.
Of course you have to weigh up what interest rate you would pay on an offset mortgage, versus what interest rate you would receive on a savings account. And it is a pain in the arse to organise remortgaging to an offset mortgage. But remortgaging is usually a good financial move anyway, if you don't want to get stuck in a high interest rate mortgage. Most people don't bother and end up paying over the odds.