The defender initially agreed a deal with former Ton boss James Grady to end his four-year spell with Airdrie United and come to Greenock.
Smyth, pictured, was then left fearing for his future when Ton chairman Douglas Rae opted for a change of manager at the end of the season.
But the Ulsterman was relieved when new boss Moore contacted him to ask about finalising his move.
He said: "Under Grady I had agreed to move at the end of the season and obviously when he lost the manager's job it fell through.
"But then Allan Moore got the job and he must have liked me as well so he approached me again. It's been ongoing for a few months."
The deal was finally completed when the 27-year-old formally signed his contract on Tuesday.
And despite enjoying his time in Lanarkshire, Smyth felt Morton's ambition and potential left him with an easy decision.
He added: "It wasn't tough to leave because at the end of the day Morton are a bigger club than Airdrie, and I want to play for a better team.
"I want to play for a team that might have a chance of winning the league and there is no reason why we can't do that.
"I'm not saying we will, but we will try, and at the end of the day that's what it's all about."
And while Smyth is eager to settle in at his new club, he has warned he will not be afraid to voice his objections should anyone let their standards drop.
He told the Telegraph: "I hope we have 11 captains. Everyone should be a leader. You want people to fall out with each other on the pitch.
"I think the manager is trying to bring in players who want to be winners. We will fall out but that's what you want.
"If you've got players who don't want to win, or who don't care about losing, then you're in trouble."
l The numbers for this week's Morton Lottery rollover jackpot of £1,500 are 2, 11, 20 and 29.
There are eight three-number winners: Liz Watson (K Beattie), E Irvine (Gibson's), J Luke (Pow), D Graham (no agent), D Mclaughlin (A Wyllie), M Stewart (D Jenkins), G Devlin (B Bradley), S Cole (B Bradley).