Okay, the whole confusion actually starts at the rally at Medda's.
There, Jack is wearing a beige shirt with a dark gray vest and a jacket. After he is arrested, both at the refuge and at Pulitzer's he still wears them.
When he agrees to become a scabber and gets his new clothes, he's obviusly given a bag for his old ones, which he has with him when Weasel brings him into the cellar with the press. (Look close, Jack's holding it in his right hand.)
There he leaves it the next day, wearing the scabber stuff while selling papers. He helps David and Sarah. After that, they go to the Jacobs' place. (This is not shown, but they must pick up Denton's article, ands I'm sure they also had some talking to do.) Jack leaves his jacket and collar there.
At Denton's, he still wears the scab trousers and shirt.
Somewhen during printing in the cellar (where he previously left the bag), he changes (telling Sarah to turn around as it is a Disney movie ;-) ) and packs the whole scabber stuff into the bag.
The rest of the movie, he's wearing the street stuff he wore at the rally, minus the vest and jacket (which are still in the bag, I'm sure).
And someone, doesn't have to be Denton, takes the bag with him to the statue (I don't think Denton did take the bag with him to Roosevelt... Most probably Jack took it himself on the wagon, we just don't see it lying there.)

Side note: Dave's blue-and-white shirt is in this bag, too! He wears it when they sneak into the cellar to print "The Newsies Banner", but someone while printing it disappears. Most probably David took it off because it was so hot down there (look how everyone's sweating) and asked Jack if he could put it into the bag. Being the nice guy he is, Jack said yes, and that means David was VERY lucky Jack did not go to Santa Fe, for otherwise he would not only have lost a friend, but also a shirt! ;-)

As you see, it all works out perfectly and no clothes do magically appear or disappear. My congratulations to Disney. :-)