Here's the concluding section of the New Exodus in Isaiah essay, which summarises the main points of the way in which the theme of the New Exodus is used in Second Isaiah:
The new exodus theme in Isaiah is predominantly a feature of Second Isaiah, where it is the controlling theme of the prophet’s message. It is
employed typologically and imminently to predict the release of Yahweh’s people from exile in Babylon, and to assist them to survey their own Heilgeschichte and so gain confidence in the progress of Yahweh’s purposes. It is also employed eschatologically through the superimposition of covenant and creation theologies to give a grand cosmic horizon for deliverance and renewal under a new and everlasting covenant with Yahweh, not only for Israel, but for all the nations. The Servant Songs of Isaiah speak of a second Moses, who will lead the nations towards this horizon. In this way, Yahweh’s future purposes for the world are telescoped into the context of the Babylonian exile. The actual return from exile did not fulfil Second Isaiah’s glorious vision; rather than the nations flocking to Jerusalem, the returnees were still subject to Persian rule and failed in their covenant obligations. However, the telescoped vision awaited its true fulfilment: ‘Just as the restoration from exile was like a second exodus, so the coming of the Messiah is like a third exodus because he has come to lead sinners – Jews and Gentiles – into the full experience of salvation’ (Waltke).