Only let the new "last month" have one extra day would fix this. That would break the Weekdays, but perhaps that's not so bad as recurring events like birthdays and holidays are then not always destined per se to be on the same weekday.
At a certain point in each year, probably at the end, we get one extra monthless day, a holiday.
Every 4 years we do two monthless days.
Also, what the fuck is wrong with Jesse? We start each month on Sunday, so that the month is divided into 4 weeks. Not one almost-week, three full weeks, and a spare hanging chad of a Sunday.