Possible Clues:

• Fish from foreign parts
• Lover of lean cuisine?
• Traps specially designed for small fish
• Traps set for little fish
• It's a little fishy the way Jack kept off grease
• Jack, perhaps, is in short, behind
• Silvery herring
• Herring type
• Argument about recipe: he and his wife had very varying dietary tastes
• Fat forgoer of rhyme