Hi YR and Welcome to the forum, you might want to have a look at the following FAQ which explains where everything is
Welcome to Benefits and Work Unlike DLA/PIP, ESA is an open-ended award. Your current award will continue until a new decision is made, however long that takes.
The 'prognosis' on DWP computer sets the new assessment date and they do tend to stick to it in the main.
So no, you don't need to do anything. Your ESA will not 'run out' if reassessment has not been completed by October. We would expect the new ESA50 to be sent out a couple of months ahead of this though.
If you want to get ahead of the game you can consider using the PDF version of the form
Using The PDF Version Of The ESA50 Questionnaire You might wish to bookmark/favourite this on your web browser now so that you can find it easily in future. This will allow you to return with further questions or comments during the ESA process without having to start a new topic each time. It helps us enormously when everything relating to the same claim is kept in one place - and I hope you will find it useful too