Tommy Sheppard, the Edinburgh East MP, became the latest high-profile figure to argue that Ms Sturgeon should not even consider whether there should be another independence vote until after the Brexit process is concluded.
But her party lost seats in Britain's national parliament in last week's national election - a result seen as a sign of falling support for Scotland going it alone.
First Minister Nicola Sturgeon has previously said a referendum could be held either towards the end of next year or the spring of the following year.
However, the First Minister has dismissed media reports, saying the SNP is still "reflecting" on the election, in which the party lost 21 MPs, and will set out its referendum plans in due course.
"It follows, therefore, that "This would mean that whilst a second referendum remains an option, the timetable gets parked".
Meanwhile, former SNP political adviser Alex Bell said the party should "shut up about indyref2", adding "the United Kingdom is in genuine crisis and there's no loss of face in letting that settle until some undetermined date".
Boxing hall of famer Lewis slams Mayweather-McGregor as 'ridiculous'
Here in Ireland, McGregor is priced at 4/1 with Paddy Power and Mayweather is at 1/6 as well. "They asked for this fight". In fact, some have argued that McGregor will have difficulty even landing on "Money", a defensive genius.
However, Mr Smith told Scotland on Sunday he was "absolutely committed to independence" despite the general election result. "It's well recognised that if we can't protect Scotland from a hard Brexit that the mandate exists to give the people of Scotland a say over their future".
There have been calls for wider consultation on Brexit in the rest of the United Kingdom, including from Britain's last prime minister, David Cameron of the centre-right Conservative party.
Mr Mundell told the BBC's Sunday Politics Scotland: "The people of Scotland sent Nicola Sturgeon and the SNP a very, very clear message in last week's General Election - with the cataclysmic performance of the SNP compared to the 2015 general election".
Ms Davidson said: "The First Minister talks about decisions made in the best interests of Scotland but doesn't she realise that is precisely what we did in 2014?"
She said: "I think what Ruth Davidson has just demonstrated there today and what she is increasingly demonstrating to the Scottish people is that she is nothing more than a one-trick pony, having to confront any issue other than an independence referendum, she is left floundering".
Has the turmoil surrounding Theresa May's electoral disaster distracted from the dramatic change in Nicola Sturgeon's fortunes?