The earlier period would place Obadiah as a contemporary of the prophet, and the latter would place Obadiah as a contemporary of the prophet Jeremiah, both of whom were prophets in the respective time periods.
The earlier period would place Obadiah as a contemporary of the prophet Elijah as reflected in 1 Kings, Chapter 18, Verses 1-16. The later date would place Obadiah as a contemporary of the prophet Jeremiah, both of whom were prophets in the respective time periods.