The 23-year-old Milne was ruled out of the World Cup before the semi-final and replaced by Matt Henry, who also played the final and has since gone on to make his test debut against England in the ongoing match at Lord's in London.
"Its been a hard time for Adam," Hesson said in a statement.
"He has worked really hard to get back to full fitness but unfortunately he is still suffering pain and is unable to bowl at his best.
"He is a young man and has a long career ahead of him so we are focussed on making sure he is 100 percent fit before he returns."
Milne will be replaced in the squad for the five matches from June 9-20 by Ben Wheeler, who played for the tourists in their warm-up match against Somerset and impressed Hesson with a five-wicket haul.
"He showed great composure against Somerset and he has a big future," said Hesson. "It will be good to have him in the squad to continue his development."
(Reporting by Greg Stutchbury in Wellington; Editing by Amlan Chakraborty)
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