When the governor enters the Smith's home (with the band playing) we see, from the inside, Ma closing the door almost shut. When the scene shifts to outside the house, Ma is again closing the same door.
Twice, Saunders says that Jefferson Smith is going to go "up" to Mt. Vernon, the home of George Washington. Mt. Vernon is approximately 15 miles south of the Capitol in Washington [you can check that using Google maps], and is right on the Potomac River downstream from Washington, so it is not "up" in a north-south sense nor in the sense of elevation. The script should have had Saunders saying that Smith was going "down" to Mt. Vernon, which is how anyone living or working in Washington would have put it.
Jeff Smith is accused of putting forward his bill on Boys Camp for his own benefit, as the supposed owner of the land on which the two contending bills lay claim. Yet if he did own this land and was concerned about selling it overpriced "at $500 an acre", he would in fact support the bill backed by the corrupt Senators, since he would make a profit from the sale of "his" land whatever the State decided to build there, Boys Camp or a dam.
When Smith arrives in Washington on the train, he's seen walking towards the exit with a porter behind him carrying his bags. The next shot shows the same porter coming into the station carrying someone else's bags.