The year used to start with spring, then counting the moons from then - except in winter, at which point nobody really seemed to care about precision.

The Ethiopian Calendar uses twelve 30 days months, plus a short 13th month to make up the difference. https://en.wikipedia.org/wiki/Ethiopian_calendar

Just start the months between days. Why do they have to align? The month is currently a completely useless measurement of time and only used to tell the approximate time of year. It doesn't matter if the next month starts half way through a day (which, of course, it does).