Neville Thurlbeck, the tabloid’s one-time chief reporter, also said that hacked messages were the source of a story about former home secretary David Blunkett having an affair.
Coulson, 47, the UK prime minister’s former director of communications, is on trial in Edinburgh accused of lying under oath in the 2010 perjury trial of ex-Socialist MSP Tommy Sheridan.
Prosecutors allege that Coulson falsely stated that, before the arrest of private investigator Glenn Mulcaire and News Of The World journalist Clive Goodman in 2006, he did not know that Mr Goodman was involved in phone hacking with Mr Mulcaire. He denies the charges.
Jurors were shown a copy of a story about the missing schoolgirl that appeared in an edition of the paper on April 14, 2002, with the headline “Milly Hoax Riddle Messages on Mobile Probed”.
The story, which appeared at a point when Milly had been missing for more than three weeks, contained precise timings and words that appeared to have been left on her phone.
Mr Thurlbeck told the court that he was looking after the newsdesk that week in the absence of Greg Miskiw, while Coulson, as deputy editor at the time, was filling in for editor Rebekah Brooks, then Wade.
He said the schoolgirl’s disappearance was “right at the top of the news agenda” and Coulson had said he wanted to make sure that the paper “was all over this story”.
Mr Thurlbeck said he had instructed Mr Mulcaire to get involved with the paper’s investigation into the Milly Dowler case and to research all possible suspects, including members of the family.