your tax disc expires at midnight on the last day of the month.Therefore you need another from midnight tonight.
There is no such thing as having a grace period,you can apply online or at post office 14 days in advance,online it says allow 5 working days to receive it ,at post office its immediate.
it is silly to leave it till last moment as thats why you get your reminder at least 14 days prior to it expiring.
Also it is an offence to not display a current car tax disc even if you do have one