The Scotland boss has also added Graeme Morrison, Andy Hall and Gregor Hayter to his squad ahead of the games against Argentina, Samoa and New Zealand next month.
But there is no place in the squad this time round for Andy Craig, Scott Gray, Andy Miller and Gordon Ross, while Ben Hinshelwood is taking a year-long break from international rugby because of a back injury.
Scotland head coach Hadden believes the performance of Godman in the European Cup win against Wasps on Sunday was the deciding factor in his selection ahead of Leeds Tykes' Ross.
Hadden said: "Phil Godman has started regularly for Edinburgh. He performed particularly well when it mattered last Sunday.
"He responded to the size of the occasion and that is something you always want to see from a stand-off.
"For me, that was a significant moment in Phil's development - to outplay completely a player of the calibre of Alex King in difficult conditions that you would think would have been alien to Phil."
But Hadden is also convinced Glasgow Warriors' Morrison can seize his chance to impress in training next week.
Hadden added: "The word 'potential' is always used when referring to Graeme Morrison and I am giving him an opportunity to start to fulfil that potential."
Morrison was delighted to have been included in the squad after only recently returning from a shoulder injury.
He said: "I thought that given my recent injuries I might not get a look-in again until the Six Nations as, until last week, I had not started a game for eight months so I will just see what happens.
"I had surgery on my shoulder and we initially thought I would be out for six weeks, but it turned out to be more complicated than first thought and turned into months rather than weeks.
"Then, during that time I had a bit of an accident as someone shut a door on my hand.
"When I first got injured I was pretty down in the dumps but it turned out to be a bit of a blessing in disguise as it meant I was able to focus on fitness and I lost some extra weight I was carrying by doing a lot of work on the bike so I am returning fitter and a few kilos lighter."
Scotland squad for training at Murrayfield from October 31 to November 2:
Backs: Blair (Edinburgh Gunners), Cusiter (Borders), Danielli (Borders), Di Rollo (Edinburgh), Godman (Edinburgh), Henderson (Glasgow Warriors), S Lamont (Northampton), R Lamont (Glasgow), MacDougall (Borders), Moffat, (Borders), Morrison (Glasgow), Parks (Glasgow), Paterson (Edinburgh), Southwell (Edinburgh), Webster (Edinburgh).
Forwards: Brown (Borders), Dearlove (Glasgow), Douglas (Borders), Ford (Borders), A Hall (Newport Gwent Dragons), D Hall (Edinburgh), Hamilton (Glasgow), Hayter (Glasgow), Hogg (Edinburgh), Jacobsen (Edinburgh), Kellock (Edinburgh), Kerr (Leeds Tykes), Lawson (Glasgow), MacLeod (Borders), S Murray (Edinburgh), Smith (Edinburgh), Taylor (Edinburgh), White (Sale Sharks).