The man came to himself and sighed. His expression was one of intense satisfaction.
The text only says that the man in the yellow suit had "intense satisfaction" one time. I do believe that he had those feelings multiple times in the book, but the narrator only tells the reader about it once.
I think that the stranger is very satisfied with himself for two reasons. The first reason that he is satisfied is because he is finally getting close to his goal of discovering that the immortal Tuck family is not a myth. He has been searching for a long time.
"During those twenty years," he said, "I worked at other things. But I couldn't forget the tune or the family that didn't grow older. They haunted my dreams. So a few months ago I left my home and I started out to look for them, following the route they were said to have taken when they left their farm."
In chapter four, the man finally hears the tune that has been haunting him for twenty years. He's intensely satisfied, because he's finally found the music box.
I think the other reason that he is very satisfied is because he believes that he is entirely in control of everything. He believes that he has the power and the plan to manipulate both the Tucks and the Fosters out of the spring. He simply believes that he is smarter than everybody else, and that he is guaranteed to win.