The Bahrain Football Association (BFA) initially said they would pursue Hudson in the courts for leaving a few months into a two-year contract but ultimately decided to focus their efforts on securing a replacement.
Hamad, Asia's coach of the year in 2004 in one of several spells in charge of his native Iraq, led Jordan to the quarter-finals of the 2011 Asian Cup in Qatar.
He also oversaw an impressive qualifying campaign for the 2014 World Cup, in which the Jordanians beat Australia and Asian champions Japan, but he turned down a contract extension before they bowed out to Uruguay in a playoff.
"Adnan Hamad has worked with Iraq and Jordan, who are teams similar to Bahrain, and has vast experience with many clubs in the gulf so we have great confidence in his ability," BFA chief Shaikh Ali bin Khalifa Al Khalifa told the body's website.
The 53-year-old Hamad, who most recently coached Emirati club Baniyas, signed a two-year deal on Tuesday.
"I am happy to work with this generation of Bahrain players who are capable of achieving historic results," Hamad said.
"There are many challenges ahead of us but I am used to this from my time with the Iraq and Jordan national teams and I know Bahrain players have strong characters and they will rise to face these challenges."
Bahrain, whose best result at an Asian Cup was a semi-final appearance in 2004 in China, face Iran, the United Arab Emirates and Qatar in first round Group C in Australia next year.
(Reporting by Nick Mulvenney, editing by Sudipto Ganguly)
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