Robinson and former Scotland international Lineen will continue with their club roles and the management team will be reviewed after the two Tests against the Pumas.
The restructuring has sidelined George Graham and Alan Tait, who assisted head coach Hadden in the World Cup and the Six Nations.
Robinson and Lineen already had a role in the national set-up in their capacity as Scotland A coaches.
They were also utilised in what Hadden termed an "observational and feedback capacity" during the week before Scotland beat England at Murrayfield - their only victory in this year's Six Nations campaign.
Hadden said in an SRU statement: "Scottish Rugby is facing some challenging games in the months ahead and I believe that now is the time to freshen up my management team.
"I am very much looking forward to working closely with Sean and Andy in the build-up and during the tour to Argentina.
"They are two top-quality coaches and there is no doubt in my mind that they will bring tremendous experience to the development of this young squad."
SRU chief executive Gordon McKie added: "Following a review of our RBS 6 Nations performances and results, the board believe that Frank is still the best man to take the national team forward as head coach and build upon the overall progress that has been made in the last two-and-a-half years or so.
"However, we also believe that our young squad is capable of making further progress with the injection of new coaching resource at this time."
Scotland avoided a second successive wooden spoon by a points difference of just three following a last-gasp defeat by Italy in their final game.
They are looking for improvement against Argentina in Rosario on June 7 and in Buenos Aires a week later.
Meanwhile, the SRU say discussions are under way with Graham and Tait over identifying other roles within the organisation.