The term ‘the Exile’ normally refers to this exile of Judaeans in Babylonia, and to the period between the fall of Jerusalem (587 BCE) and the conquest of Babylon by Cyrus of Persia in 539 BCE, when Jews began to be permitted to return home, according to Ezra 1.
The Israelites deported by the Assyrians are not heard of again (hence the legend of the ‘Ten Lost Tribes’; in actuality most of the people continued to live in their homeland). Those deported by the Babylonians became a flourishing community, and remained so until 1948 CE. Many of them, however, returned to their homeland at various times: see the books of Ezra and Nehemiah. It is likely that these returned exiles were key players in the editing of most of the books of the Old Testament and the establishment of the Jewish religion.
By the process of restoration, post-exilic Jewry was given hope and directives to regain their relationship with God by righteous living via the various prophecies, often in the form of eschatology.