Morris, 24, was released by the Iron in the summer, having spent the season on loan at Chesterfield.
The former Republic of Ireland U21 international came through the tanks at Leeds United before moving to Scunthorpe in 2006.
"He is the missing piece of the jigsaw that I've been talking about all the way through," said Gulls manager Martin Ling.
"He's a left-footed player who is happy playing out wide. We've got good wide players here but we do need a left-sided player, and Ian is very comfortable out wide. He can play in the middle as well but has played most his career out wide.
"He has got Championship games behind him and has good pedigree. In previous times, Ian wouldn't be dropping down to League Two because there would be clubs out there for him.
"In the current climate, he has dropped down to find his home and I'm very pleased. When I saw his name on the available list, I was immediately hoping we had enough money to do the deal.
"We have a group of players at the training ground that want to learn, want to be successful and want to work hard. Ian will not disrupt that approach and that's important.
"We've had to act quickly because he's been on trial with Hereford, and scored two goals as well, so we're very happy."