By default, customers have an option to start a new chat directly from the previous chat via the 'Start chat again' function:
But there is an option in the chat button settings that allows you to disable this function.
First, navigate to Configuration > Chat > Chat Buttons > Edit the button you want to use.
Then click on "Chat Window" on the left side of the options, and you'll see a check box for "Hide start chat again button"
Now the visitors will not be able to start a new chat from the old one: