The Scottish Premier League will meet again to reconsider Inverness Caley Thistle's promotion bid.
On Tuesday, the First Division champions did not secure the necessary 8-4 majority from top flight clubs.
But a general meeting has been called for 22 June after Hibs and Hearts asked that the issue be discussed again.
Livingston and Dundee, who did not support the Highlanders' groundshare proposals, have suggested that they may well have a change of heart.
Caley Thistle, who want ground-share with Aberdeen because their stadium does not meet SPL requirements, received seven votes on Tuesday, but they immediately appealed against the decision to deny them promotion.
The Scottish Football Association said they will hear the appeal before the end of June but that may now be avoided.
If the second vote goes the way of Inverness, Partick Thistle will be relegated - but they will not go down without a fight.
The Jags, who have previously threatened legal action against the SPL, have already issued a demand for £1m in compensation if they are demoted.
Pearse Flynn, whose Lionheart consortium have secured an agreement to take over the running of Livingston, has confirmed he is backing Inverness.
"I've no axe to grind but the key point is do you believe in promotion and relegation, and I do," said Flynn.
However, Livingston's vote will be cast by the club's administrator as Flynn does not take charge until 1 July.
"It would be consultation between us and Kroll but I think they would listen to us because we are the guys taking it forward," added Flynn.
Despite suggestions that Dundee were also prepared to change their vote, a Dens Park spokesman said: "Our position and our view of this matter has not changed.
"If the SPL or the clubs who voted in favour of Inverness can demonstrate to us that a vote to admit Caley to the league is not in contravention of the rules and therefore not likely to be open to successful legal challenge, we would be prepared to consider the matter again."
Hibs managing director Rod Petrie has suggested that Dundee had mistakenly believed Caley had missed a 31 March deadline for their ground-sharing proposals.
Inverness director of football Graeme Bennett is hopeful of winning the second vote.
"We are quite confident, although I was quite confident before the first one," he said.
Meanwhile, Dundee United are set to invite Caley Thistle to meet them privately before the new vote.
A Tannadice spokesman said: "In light of the SPL decision to reconsider the Inverness Caley situation, we'll again look at all aspects of the situation, especially if additional valid information is brought to the table."
In summary, Inverness Caley won the first division, Normally that would mean that they get promotion to the SPL. However, they don't have the stadium in place yet to match the SPL's standards. That meant Partick Thistle stayed in the SPL rather than get relegated.
Inverness Caley offered to play at Aberdeens stadium, a two hundred mile round trip from their own stadium. They needed 8 votes for this to happen, they only got 7