You can start the new sensor straight away.
BUT ... I always need time for my sensors to bed in. If I started my sensor as soon as I inserted them, I would not be able to rely on any numbers or the first 24 hours so would be finger pricking anyway. I prefer to wait the 24 hours when a sensors falls off.
On the other had, if my next sensor was due to start today, I would have one ready and willing.
But the short answer is you do not have to wait until the last sensor was supposed to run out before starting a new one.