The Killie defender started on the bench in last week's opening 1-0 win over Hibs as Killie manager Jim Jefferies opted for Ryan O'Leary and Simon Ford as his centre-half partnership.
Wright, however, was called into action during the first half of the clash as O'Leary had to go off after dislocating his shoulder.
Wright knows there is intense competition to be in Killie's starting line-up this season after Jefferies strengthened his squad over the summer.
Having got back in through O'Leary's misfortune last weekend, Wright is determined to stay there.
He said: "It was disappointing not to start as you're always looking to be in the team for the first game of the season.
"But the manager said it's going to be a long season and to keep my head up.
"But I've been lucky and Ryan O'Leary has been unlucky and I've managed to get back in the team and hopefully I'll stay there.
"There is strong competition all over the squad.
"We have four central defenders competing for the two places, the manager has just signed Donovan on loan so we have a few strikers and there are loads of midfielders.
"So there is good competition."
Killie take on St Mirren at Love Street and Wright is confident of building on last weekend's winning start.
But he knows Killie face a tough test against a Saints side which ran SPL champions Celtic close last weekend.
He said: "It'll be a hard game but looking at the results from last weekend, Inverness winning, Hamilton winning and St Mirren playing well, it's going to be a hard league this year and you won't get an easy game this season.
"It's important to get a good start and for us to do that in our first game was good and that sets us up nicely.
"We've now got St Mirren away and then Hamilton at home, so it would be good to pick up points from them."