The Scottish Government has agreed that a legal requirement to force local authorities to introduce the policy from August will be put off until November.
The Government was warned earlier this month that not all councils would be able to deliver 600 hours of free childcare to vulnerable two-year-olds from August 1.
David O'Neill, president of council umbrella body Cosla, welcomed the concession.
He said: "Local government has risen to the challenge this policy has presented and I'm pleased the Scottish Government is supporting councils during the early stages of implementation in such a constructive way. Shifting the start time of the legal duty.... responds to what councils are telling us that they don't want to be penalised, despite all the effort that has been put in."
Cosla said councils would be offering places to families from the start of term in August, but the extra time allowed "flexibility" to resolve any challenges over capacity.
Education Secretary Michael Russell said: "We are at the start of a journey which will see significant further expansion of early learning and childcare provision to help give our children the best start in life and make Scotland the best place in the world to grow up."
However, Kezia Dugdale, Scottish Labour's education spokesman, described the move as a policy "collapse". She said: "The Scottish Government has had to admit that they cannot deliver the uncosted childcare sweetener they said they would deliver before the referendum. If they cannot deliver what they said they would now, how can we possibly believe any of their other uncosted plans for the future?"
Liz Smith, young people spokeswoman for the Scottish Conservatives, said the flagship policy was in "disarray".
She said: "Parents across Scotland who were promised this need to know if these facilities are going to be up to scratch for everyone, and why some children may be made to wait nearly three months longer than pledged.
"Given the disingenuous nature of the announcement, the Scottish Government must give a formal statement to parliament as soon as possible."
The concession comes after a major row earlier this month between the Scottish Government and Cosla over how much the scheme will cost. Ministers argue the move to deliver 600 hours of childcare to two-year-olds from workless households will require capital funding of about £61 million, but Cosla argues the real figure is closer to £114m.