Other mistake: When Charles takes a horse to chase after Carrie in the hot air balloon, they use a stunt double. He's not even dressed like Charles. Then a few minutes later, they use the exact same stunt man to jump on the hot air balloon. Just looks silly.
Revealing mistake: When Charles steals a horse to chase down the balloon with Carrie in it, you can see it's a stunt double. Charles is wearing his hat, beige shirt, and his knee-high boots with pants tucked in. The stuntman has a pink shirt, no hat, and long pants with shoes. Maybe even white socks.
Factual error: When the man is holding onto the rope from the balloon, you can see modern tire tracks in the field.
Factual error: When the hot air balloon comes down with Carrie in it you can see N410AA printed on it. This is commonly called a tail number on planes and is issued through the FAA, which of course did not exist during the time period of the episode.
Revealing mistake: During the scenes when they are chasing the balloon in the prairie, the tire tracks from the camera car are easily visible.
Suggested correction: Wasn't Mary the one in the balloon?
Mary was in the balloon earlier in the episode with the man who runs it but near the end, Carrie wondered off into it and it took off with her inside ALONE.