McGregor, who left St Mirren at the end of last season, completed a dream move to the Ibrox club in June.
But the centre-half only realised what he had done when he was heading to a pre-season game with Buckie Thistle last month.
He said: "I didn't know that before every game they put on 'Simply The Best' before the bus pulls in to get everyone going.
"I was half sleeping and woke up to 'Simply The Best'. It was a surreal moment.
"That was the moment I thought: 'I'm a Rangers player now!' Hopefully I can make a difference here and be remembered in a good light."
He added: "The fact I am now a Rangers player hasn't really been at the forefront of my mind.
"If I thought about the magnitude of everything it would be too much to digest.
"I put the blinkers on and focus on what I can bring to the team.
"We have a great set of guys with no egos. No-one is big time about anything. It's a great environment to be in. So you blend in with that and get on with it.
"You just want to win games of football. The level of the club and the expectation levels are higher, but if we can get on with winning we'll keep everyone happy."
McGregor was a boyhood Hibs fan, but will be desperate to beat the capital club if he plays in the Petrofac Training Cup match at Ibrox tonight.
He said: "I am from Leith so I grew up a Hibs fan. But I hope we do well against them.
"This game could help us set a benchmark for the season if we can perform well."