Lessons have been learned by the company which took over the running of the prison escort service, according to its Scottish operations director.
Campbell O'Connell is confident the contract will be rolled out
Campbell O'Connell said Reliance Custodial Services was confident that the next phase of its £126m contract will be approved next month.
Its extension was put on hold after a series of high-profile errors.
The Scottish Executive said the company's hopes of rolling out the contract were "premature".
It wants weeks of good performance before giving the go-ahead.
Mr O'Connell said the firm had overcome the problems which blighted the first few weeks of the contract.
He said Reliance was now transporting more than 90% of prisoners to court on time.
"I am confident in what we are achieving here and we are providing a good service," he said during a media open day at the firm's control centre in East Kilbride.
The company escorts prisoners to and from court
Mr O'Connell said the company had to meet certain criteria before the programme could be rolled out to its second phase - covering Dumfries and Galloway, central Scotland and the rest of the Strathclyde force area.
But he said he was confident it would be approved and would win the backing of the Scottish Parliament's justice committee on 8 June.
Reliance took over the contract for the west of Scotland last month.
Mr O'Connell admitted that it was "unfortunate" that a prisoner was released in error during the first week.
"The circumstances have been reported to the police and there is an investigation.
"Through time we will know what happened," he said.
The third phase of the contract covers Tayside, Fife and Lothian and Borders, with the fourth taking in the Grampian and Northern force areas.
The final phase involves non-court work such as
inter-prison transfers and warrants.
Mr O'Connell added: "I am confident that as we move onto other areas the transition will be a lot easier.
Reliance took over the contract last month
"There are a lot of lessons that have been learned."
However, Scottish National Party justice spokeswoman Nicola Sturgeon said it was premature to be talking about the roll-out of the contract.
"Reliance assurances have been shown not to be worth the paper they are written on," she said.
"They said when they took over the contract that they were up to job - they weren't.
"We don't now need more promises, we need to see hard evidence of consistently good performance.
"Until that has been delivered over a period of months, there should be no talk of rolling out the contract."