The poem is built like a proverb, the kind of short saying that sounds like old wisdom, but it twists the form in a way that makes you stop and think. The first three lines are puzzling on purpose. A "chin of good cheer" suggests someone holding their head up and putting on a brave face. Set against a famine, though, that cheerfulness is called "the greatest extremity." One way to read this is that forced optimism in the middle of real suffering is a kind of excess, maybe even a strain or a denial, as extreme as hunger itself.
The second stanza shifts the tone. "Some part of this day / is another" is quiet and open-ended. It could mean that every day carries pieces of other days, or that hard times and easy times are mixed together. The speaker doesn't seem to preach. The mood is dry, thoughtful and a little uneasy, like a riddle that you only half solve and keep turning over.