Ally McCoist's side were convincing 8-1 winners over the Bully Wee at Ibrox on Monday night as Kris Boyd fired a hat-trick, Lewis Macleod and Lee McCulloch got doubles and Fraser Aird scored on his return to the first team.
The Light Blue legions had voiced their fury with below-par performances against Hibernian, Hearts and Falkirk.
And McGregor knows the Gers must raise the bar again to thrive under the Ibrox pressure.
He said: "I have noticed since I've come here that the expectations of the fans and the standards they demand are so high.
"It does put added pressure on us but that's why we are at this club, to deal with that kind of thing and deliver for them.
"We are all looking forward to performing every week and hopefully giving the fans their money's worth. We need to keep standards up now.
"Everybody wore a smile on their face after the Clyde game and the only sticking point was the goal we lost at the end of the game.
"It was a disappointment to lose a clean sheet but overall we did well as a team and we have to move on and keep that going.
"The good thing is we've not reached our peak yet and I think there is still a lot more to come from us."
Rangers return to Championship action this weekend when they face Dumbarton.
McCoist's side got their first league points last Friday night against Falkirk and McGregor is determined to keep moving in the right direction.
"It was great to go out at Ibrox against Clyde and finally give a good team performance," he told rangers.co.uk.
"There were a few good individual performances too. Some of the boys were excellent and to score eight goals against any team is a great achievement.
"It will definitely raise confidence levels. That's what football is all about.
"Obviously we know we were playing against part-time opposition from two leagues below us but we still had to put the ball in the back of the net.
"There are boys who will have taken a lot from Tuesday. We're hoping now that the likes of Kris can kick on.
"He can get us 30-plus goals this season."