The 34-year-old is on course to be the party's new Westminster leader, thrusting him into the political spotlight just three years after he first entered the Commons as part of the 2019 intake.
He is widely expected to replace Ian Blackford, who announced he will stand down at the group's annual general meeting on Tuesday. Insiders said they would be "highly surprised" if any other candidate put their name forward.
Mr Flynn, a former councillor in Aberdeen, has been seen as potential leadership material for a number of months. But the speed of his rise took many by surprise.
"He's seen as a good option because people like him," said one well-connected party figure. "He's effective in the chamber. Obviously the proof in the pudding is whether he's effective in holding that group together."
It is no secret some SNP MPs feel dissatisfied and detached from the party leadership in Edinburgh. There is a sense of frustration and boredom. Mr Blackford's handling of sexual harassment claims against former chief whip Patrick Grady also caused unhappiness earlier this year.
"Ian hasn't been landing a blow on the UK Government for some time now," the party figure said, echoing wider concerns about the Ross, Skye and Lochaber MP’s performance in the weekly Prime Minister's Questions bear pit.
The sacking of Joanna Cherry, the high-profile Edinburgh South West MP, raised eyebrows last year. Outspoken and frequently at odds with the party leadership, particularly over transgender issues, Ms Cherry is nevertheless a formidable talent. She is also seen as an ally of former first minister Alex Salmond, whose Alba Party attracted defections from ex-SNP MPs Kenny MacAskill and Neale Hanvey.
"I don't think Ian could have done anything differently that would have kept Kenny or Neale in the party, but some leaders – let's say Alex Salmond, for example, was very effective at keeping people who had been his archnemeses within the SNP, people that he didn't necessarily see eye to eye with, like [former health secretary] Alex Neil," the well-connected figure said. "He put him in his Cabinet, because it's a better way to deal with them. Ian probably was too eager to reward loyalty with frontbench positions."
When news of Mr Blackford's resignation broke, Ms Cherry did not hide her feelings. “I’m pleased to hear this,” she tweeted. “It’s time for fresh leadership and tolerance of debate and diverse viewpoints."
Ms Cherry ran against Mr Blackford for the Westminster role in 2017, but has ruled herself out this time round. However, she is widely expected to make a return to the frontbench under Mr Flynn, with allies already said to be spreading word that a job is in the bag.
Edinburgh East MP Tommy Sheppard, who canvassed support in 2017 before withdrawing from the race, also confirmed he has no plans to throw his hat in the ring, joking: "If I was ten years younger, I might think about it." He said the new group leader "will want to review our tactics and strategy" and there is an appetite for an "open conversation" about the route forward.
"This is as good a time to make a change as any," he told The Scotsman. "We all need to have a rethink and a reset around the Supreme Court judgment anyway."
The well-connected party source was blunt about the situation. "I think there needs to be a serious discussion with the new leader and the party leadership at Holyrood about what the role is for the Westminster group in pushing the independence argument forward, because it's all very well and good to stand up and make it every Wednesday [at PMQs] as Ian Blackford has for the last year, but the whole chamber groans when he gets to his feet," they said.
"The Prime Minister knows exactly what he's going to ask. He gets shot down in two sentences’ worth of answers from the PM, and that's it. It gains nothing."
There had been persistent rumours about Mr Blackford's position. Insiders said he mentally "crossed the rubicon" in the past couple of weeks. Reports of a leadership bid by Mr Flynn surfaced two weeks ago, and the coming AGM brought things to a head.
Mr Blackford was close to Nicola Sturgeon and unfailingly loyal. His exit will be a blow to the First Minister, and a change of leadership in Westminster could bring a different dynamic. For one thing, Mr Flynn is seen as more supportive of the oil and gas industry.
"Stephen has certainly got what it takes in terms of him – Stephen the person is leadership material," the well-connected figure said. "But the tricky thing is, who's the team around him and what are they going to do? He's not saying anything in advance of next week, when we know whether he's the new leader or not.
"He's pretty pragmatic, he's pretty sensible. I think he will give a bit more punch to some of the constitutional issues, social issues. It's vital that he picks a good team from the get-go, though."
They argued the SNP’s Westminster frontbench should be a "more fleet of foot fighting force", insisting: "Using every single person within the group to shadow every single government role and heaps more besides – what's the point? Have somebody who can talk on the economy, have somebody who can talk on defence, have somebody who can talk on the key issues."
With Ms Sturgeon seeking to turn the next general election into a “de-facto” referendum, the SNP’s new Westminster leader will want to make sure they are well placed to play their part. Mr Flynn could have his work cut out.