The party has, for the first time in four decades, lost overall control of Glasgow Council. They came second in the election and the SNP came first.
The Scottish National party leader said that although the Conservatives did well "by their standards", they did so by taking votes from Labour, not by eating into SNP support. But both were outstripped by the SNP, which added 31 councillors to win a total of 431 seats.
"SNP councillors and SNP councils will put their communities and the people of Scotland first". Scottish Conservatives even have their first ever councillor in the Western Isles.
Ms Hair also increased the party's share of the vote by 15% in Angus South in the Scottish Parliament election a year ago. Over one in ten votes on May 4 were cast for independent candidates - in the Highlands and the Islands in particular, council elections are often still genuinely local rather than partisan battles.
Her bullish stance was echoed by SNP number-crunchers who last night claimed they had attracted 105,000 more votes than at the last council elections in 2012.
Last week the UK Prime Minister accused the European Union of interfering in the upcoming UK General Elections on 8 June, causing more tensions with the 27 member block and drawing criticism for provoking what some analysts fear could be a "no deal Brexit" situation.
It did not, though it did become the largest party in each, as it now is in half of Scotland's 32 councils.
He said it would be a "major moment" for the SNP if it wins power in the "iconic" city of Glasgow, which has been part of Labour's traditional Scottish heartlands. Ukip's sole win was a gain from Labour in Lancashire, as its vote share collapsed nationally from from 22% to less than 5%.More news: West Ham 1 Tottenham 0: Eric Dier makes key title race admission
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Labour leader Kezia Dugdale described the results as " obviously a disappointing election".
John Curtice, professor of politics at Strathclyde University and one of the most respected figures in analysing Scottish election results, said the party could struggle significantly more in the region than elsewhere.
Scottish Labour deputy leader Alex Rowley meanwhile insisted his party is still a "major party" in local government, despite losing seats.
Launching his campaign to be re-elected as an MP, Salmond yesterday warned the Tories not to make "presumptuous announcements".
But, equally, it could be a sign that the party has lost ground.
The Scottish Greens will be happy with their showing in last week's election.
Four councils that quit the Convention of Scottish Local Authorities (COSLA) in a dispute over decision-making processes - Glasgow, Aberdeen, Renfrewshire and South Lanarkshire - are expected to rejoin at the COSLA convention at the end of June.
She continued: "This city is bursting with potential and there is really an opportunity now for a bright new start and administration that will see Glasgow fulfil its potential".