Ove, of the Swedish-Walloon Descendents group, has sent me a 600 page document, and from this I glean that the earliest Swedish Söderman in our line appears to be Johan Ersson, born 1685. So Johan's father was Erik Jansson, born ‘about’ 1655. They don’t give where either of them was born, though they died in Forsmark.
Johan Ersson married Elisabet Bovin, whose father (Tussin, or Toussaint, Boveng/Bovin) is named in the above screenshot on the same entry. He is the father of Tussin Söderman, born 1723.
When I was in Slovenia at a permaculture conference in 2012 I roomed with Kiki, a woman from Lund in Sweden, and, jumping on the fact that her husband did a lot of family history, I asked her about the name "Sederman".
She was of the opinion that it might have been Finnish, as in Cedar Man, ie. a woodsman who worked in cedar forests, and that the family might have moved from there to Sweden. Although we didn’t know about Söderman at the time, she also linked it to Södermanland, “men from the south” and she said that rural Swedes were 'parochial' enough to consider someone who came from the next village to the south of their own, as an outsider, and would often nickname an incomer as that… “you know, that Johan, the söderman”!
That’s possibly how the ‘extra’ name got put on the end of the patronymic Johan Ersson, if he came from the south - Södermanland even? - to work in Forsmark.
Jan Olsson is the ‘keeper’ of this info, so I am hoping to hear back from him...