McAllister said the candidate for next March's election need not be European but must be capable of defeating Infantino, uniting football's confederations and delivering lasting governance reform.
Speaking to Reuters at the World Football Summit in Madrid after UEFA's Executive Committee met in Thessaloniki, she rejected suggestions that opposition to Infantino had lost momentum despite the FIFA president appearing to retain enough support to win another term.
UEFA's efforts were not about installing a European figurehead to lead a rebellion, she said, but building support for a candidate with a clear reform programme.
"I don't think so," McAllister said when asked whether Infantino's opponents had overestimated the damage to his authority.
"We always knew that things wouldn't just change overnight. That would be naive. We knew that there would be a resistance to change as well, of course.
"We're taking things step by step, both legally in terms of the political context with an alternative candidate and in terms of the programme of governance reforms that we need to see."
UEFA 'united and very concerted'
Swiss Italian Infantino, 56, faced the biggest challenge to his presidency following the collapse of his proposal to sell a stake of up to 20% in FIFA competitions, including the World Cup, to private investors.
The plan prompted fierce opposition, with UEFA, the Asian Football Confederation and CONCACAF among those calling for change at the top of FIFA. Several European national associations have since withdrawn their support for Infantino's re-election, with Greece becoming the latest on Tuesday.
McAllister said UEFA nations remained "united and very concerted" in their determination to change the way FIFA is governed, but stressed that the dispute went beyond Infantino.
"This was always about more than one person," she said.
McAllister said UEFA had been working quietly on several fronts and regarded the reform platform as more important than the identity or nationality of the eventual candidate.
"There's lots of prongs to any campaign to change something," she said. "There are tactics that you need to follow, but I feel very reassured that we're on track, that we will try to find a candidate that can unite all of the confederations, because it's not just UEFA; all of the confederations feel strongly about the need for change.
"And that candidate would then stand on a manifesto of reform and modernisation."
Infantino has announced that he will seek another term at the election in March. McAllister said his opponents must find a candidate capable of doing more than mounting a symbolic challenge.
"If you're in a position where you know the incumbent will be standing again, and Gianni Infantino has made it clear that he will be a candidate in the election unless something changes in the time before next March, then it's incumbent on those who oppose him to get the right candidate," she said.
"You don't fight an election unless you can win that election."
Not insisting on European challenger
McAllister said UEFA would not insist on a European challenger if a credible reform candidate emerged from another confederation.
"This is about the programme of change, and I think we have to get that right first," she said.
"We have to have a really clear template of what good governance looks like in global football, and it isn't what we've seen from FIFA over recent years."
McAllister said the reform programme should include greater accountability and transparency, proper recognition of conflicts of interest, stronger protection for whistleblowers and more diverse leadership.
Asked whether UEFA could maintain a working relationship with Infantino if he were re-elected, McAllister said European football would remain a powerful force in the global game, but that FIFA's leadership should learn from what has happened.
"That's not our problem as much as it's his problem," she said when asked what Infantino would have to do to restore UEFA's confidence.
"If he were to be elected, one would hope that he would have learned the lessons of a period that the game has seen as a real existential crisis."
Neither FIFA nor Infantino responded immediately to a Reuters request for comment.
McAllister said UEFA would focus on making the case for reform rather than looking too far ahead before nominations had opened.
"The priority for us now has to be about convincing the world of football that it belongs to them," she said, "but that we need cultural change in the way in which the game is governed."