The 25-year-old scored five goals in August and he matched that tally last month after netting a hat-trick in a 4-1 victory over Walsall.
"I'm really delighted with it and didn't see it coming," said Morris. "To win it two months in a row is massive and again thanks have to go to the lads and the manager.
"All I was aiming for after last month's award was to work hard for the team and see what happened. It's nice to win it again, but to be top of the league is the main thing and carrying on our performances.
"I'll always try my best and again we see what happens after that. Anything can happen, so we just wait and see."
Scunthorpe manager Graham Alexander added: "It's great recognition of Josh's and the team's performance. He's continued his form from the first day of the season starting and it is another acknowledgement of what he has been doing.
"Every credit to him. He fully deserves it. He won this award in August, but I'd say his performances in September have been even better and it's a great one for Josh to win. It's a great little nod to the team and how the team are doing.
"His performances have been that high, he's been impossible to ignore. As you saw from gaining the first award, it doesn't make him think any different about his performances or himself, he continues to work hard for the team and that's the basis of him winning this award. He's a very calm individual and very mature. He knows what he needs to do be in the team and that the team comes first. His workrate for the team is always top notch."
Morris joined Scunthorpe on a free transfer in the summer after leaving Bradford City.