The Saints were linked to defensive end Ziggy Ansah early in free agency, but Ansah wound up signing with the Seahawks a couple of months ago.
Hood split last season between Washington and Miami. He appeared in 13 games and made nine tackles.
Hood was a 2009 first-round pick of the Steelers and spent five seasons in Pittsburgh before moving on to the Jaguars as a free agent in 2014. He moved on to the Bears for the start of the 2015 season and finished it in Washington.
The Saints have Sheldon Rankins on the PUP list to open camp as a result of his torn Achilles and David Onyemata will be suspended for the first game of the year, so Hood could find his way into the rotation up front with a strong summer.