Factual error: The Marine Colonel in Supergirl played by Eddie McClintock spends all of his time wearing Marine Officer's Dress Blue Bravos. Marine Dress Blues are ceremonial uniforms. On rare occasions, the Charlies or Deltas (no coat) may be worn as a work uniforms (the primary exception to this rareness being recruiters). The full uniform is certainly not worn for day to day work in the field.

Supergirl (2015)
1 factual error in Manhunter
Starring: Calista Flockhart, Melissa Benoist, Mehcad Brooks, David Harewood
Continuity mistake: Rick Malverne has handcuffs on when J'onn goes in as his dad. When he figures out it was J'onn, he tells them they have "23 hours and 14 minutes," looks at his watch and the cuffs are gone. (00:21:30 - 00:25:00)
The Adventures of Supergirl - S2-E1
Trivia: When Cat Grant walks to her office door, she screams out, "Miss Teschmacher!" In Superman and Superman 2, Miss Teschmacher was Lex Luthor's assistant and him screaming her name was his way of getting her to run to him.
Question: Why did they change her costume and hair style?
Answer: I can't say for certain, so I'm surmising that the change was to "catch up" with the other Arrowverse shows, particularly Arrow and The Flash, whose lead characters have undergone various upgrades and redesigns of their superhero suits over the years; after four seasons, Kara's probably just felt a little stale. And I believe the change in hairstyle was Melissa Benoist's personal decision and had nothing to do with the character.
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Answer: Another part of why the change was that it's cold in Canada in the winter and the old costume didn't protect her from it, whereas the new one is nice and warm.