Bereavement benefit coming to an end can I submit esa claim now?

My MIL (61) has been on bereavement benefit for almost a year, it is due to finish mid June but due to health problems she will not be fit for work. Can she submit her esa claim before the bereavement benefit ends so that she is not without money while it is processed?

Been on hold now to esa for just over half an hour waiting for someone to answer.

Thanks, grateful for any advice.