The Stoic shift is about a shift in what we value. Rather than valuing attainments, possessions and pleasures in the external world, we value the development of character.
Character means growing into ourselves, so that we are less dependent for our contentment on external factors. We become happier about facing a setback, and being unperturbed by it, than about a success in the world. Indeed, a success that captures us with pleasure or delight is a deficit. Whereas a setback that leaves us unmoved is a true gain, because it represents a gain in character.
This has clear parallels with Zen and Buddhism, even if the methods differ. According to Jeff Shore, a Rinzai Zen teacher, the Rinzai ideal is a person who is simply dependent on nothing at all.