Father And Son, which has been screened on the last three Monday nights, has also been within the Top 10 most popular programmes of the day on any television channel.
Its first episode pulled in 353,000 viewers -- or a 26pc share of the audience.
The following Monday, on July 6, the drama was watched by some 366,000 viewers, or a 27pc share of the audience.
The audience for the series was expected to increase further last night for the third instalment and Montrose chiefs expect the drama to finish on a high.
The four-part drama stars Scott as a reformed criminal forced to confront his past.
The cast also features Stephen Rea, Sophie Okonedo, John Kavanagh and Flora Montgomery. The script was written by Emmy Award-winning Irish actor Frank Deasy.
Father and Son was meant to be filmed in Manchester, where much of it is set.
However, this schedule changed when the Irish Film Board intervened and much of the series was shot mainly in Dublin.
Scott plays ex-criminal Michael O'Connor, who returns from his quiet life in Ireland to Manchester to save his son Sean from a life in prison. He is joined by his pregnant girlfriend Anna, played by Flora Montgomery. He is also seeking to redeem his own troubled past.
Sean is sharing a cell with Barrington Smith (Terence Maynard), Michael's old partner in crime.
Barrington's kidneys are failing and he needs to get out of jail so he can get an operation to save his life. Barrington believes Michael is the only person who can rescue him.
Forced to make a deal with Barrington, Michael is reacquainted with the ghosts of his past, including Augustine Conlon (Stephen Rea), an IRA gun dealer turned property magnate, and Greater Manchester Police DCI Tony Conroy (Ian Hart). Conroy has been pursuing Michael for years and is determined to put him behind bars for good.
Jane Gogan, commissioning editor for RTE Drama, described Father and Son as a rich and complex high-end drama.
"RTE's association with ITV in commissioning this series is an important development. While much of the drama was based in Manchester, it was filmed mainly in Dublin, creating revenue and employment for the Irish television industry," she said.