Scottish Labour leader Kezia Dugdale said the party had run the SNP so close it could win any seat in the next election.
Ms Dugdale also hailed its six new MPs for providing nearly one quarter of the resurgent party’s gains at Westminster.
She described the increase from one seat in 2015 as a “fantastic result and hugely encouraging”.
Ms Dugdale said Labour had also run the SNP “incredibly close” in several seats, coming within 100 votes of victory.
She said: “Every single seat is in contention next time round.
“That’s tremendous progress for the Scottish Labour Party.”
Close calls included being 60 votes behind the SNP in Glasgow South West and 75 votes short in Glasgow East.
Ms Dugdale said her priority was for the SNP to shelve a second independence referendum.
She also said Theresa May was making a “grave mistake in trying to plough ahead to form a government”.
She said Mrs May should instead resign, to give Labour an opportunity to form a minority government.
Jeremy Corbyn becoming Prime Minister was “still a possibility”.
However, Ms Dugdale ruled out any deal with the SNP or other parties.
She said: “There will be no progressive alliance.”
She said she was “thrilled” for the six new MPs, who will join sole incumbent Ian Murray, who increased Labour’s vote in Edinburgh South by nearly 16 per cent.
New Glasgow North East MP Paul Sweeney, a community worker, achieved Labour’s second biggest swing north of the Border, recapturing what was once Scotland’s safest seat from the SNP with a 9.2 per cent increase in its share of the vote, albeit with a majority of just 242.
Mr Sweeney said it had seemed too close to call even at the start of the count, after expecting during the campaign to only be able to reduce the SNP’s 9,222 majority.
He said voters had been attracted by Labour’s manifesto, such as ending austerity and a real living wage.
When he saw many Labour posters and few SNP ones in Alexandra Parade - one of the main roads in the constituency - “I thought something was happening”.
In the other gains, all from the Nationalists, North Lanarkshire councillor Hugh Gaffney won Coatbridge, Chryston and Bellshill with an 8.7 per cent vote increase and an 1,586 majority.
South Lanarkshire councillor Ged Killen took Rutherglen and Hamilton West with a 265 majority and 2.3 per cent increase in the vote.
In the east, Shelter Scotland staffer Danielle Rowley, 27, who is also daughter of Scottish Labour deputy leader Alex Rowley, became the party’s youngest Scottish MP, winning Midlothian with an 885 majority and 6.2 per cent vote increase.
Prestonpans primary school teacher Martin Whitfield took East Lothian with a 5.1 per cent increase and 3,083 majority.
Fife Council deputy leader Lesley Laird won Gordon Brown’s old seat of Kirkcaldy and Cowdenbeath with a 259 majority and 3.5 per cent increase.