Re: Hardly rocket science
You can (and pretty much always have been able to) renew your tax in advance of the expiry date - I don't really understand why this month is any different from that perspective.
As far as tax starting on the first of the month - I can see how this made sense in the age of the multi-coloured paper-based tax disc, but under the new regime it seems unnecessary. Maybe it will transition to this in the future once the populace has got used to the idea of no paper tax discs,