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Morgan, 25, made three appearances in the Europa League for the Reds having impressed Brendan Rodgers on a tour of the USA in 2012.
After leaving Liverpool in 2014, he joined Yeovil Town, before spells at Accrington Stanley, Hemel Hempstead Town, Colwyn Bay, Halifax Town and Sligo Rovers.
He scored a hat-trick for Curzon in their televised FA Cup round defeat to AFC Wimbledon.
Meanwhile, striker Astley Mulholland suffered a broken leg in the friendly win at Abbey Hey on Tuesday night.
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