Membership of the party now stands at more than 57,000, a huge surge from the 25,642 recorded last Thursday when a majority of Scots voted to remain part of the UK.
The party says its membership now outstrips that of the Liberal Democrats across the UK, making it the third largest political party.
SNP business convener Derek Mackay said the influx of new members “shows no signs of slowing down”.
He seized on comments made by former Labour first minister Henry McLeish that his party was “not in good shape” in Scotland.
Mr McLeish told the Daily Record: “We need to get our values in order and explain what they are. Labour needs a review, not of leadership, but of the wider movement.”
Mr Mackay said: “What a contrast with the No campaign parties in the last few days - we have overtaken the Lib Dems at UK level and Labour are in complete disarray in Scotland.
“As the whispering campaign against Johann Lamont continues apace at Labour conference, the warning from Henry McLeish that Labour is in danger of dying out is yet another blow to her floundering leadership.
“But Henry McLeish is absolutely right - Labour’s position in Scotland is growing more precarious by the day as more and more people in traditional Labour heartlands switch to the SNP.
“It is clearly the SNP that the people of Scotland trust to stand up for Scotland’s interests - and to hold the Westminster establishment to account on their vow of substantial more powers for Scotland.
“This new democratic movement in Scotland won’t accept the same old politics as usual from Westminster.”
Line of Duty star Martin Compston announced on Twitter that both he and his mother had joined the SNP following last week’s referendum.
The actor, who had been involved in the campaign for independence, said: “The referendum was never about parties for me, but now the dust has settled I believe the best way to move forward and get the change we need is the SNP.
“Myself and mum just joined the tens of thousands who have signed up since Friday and continue to believe in hope over fear.”