Continuity mistake: As they are fleeing the alien base, Supergirl and the others run towards the portal, Winn activates it and jumps through. In a following shot (after a 'friendly' alien helps them), he is standing behind Supergirl again. (00:32:00 - 00:32:35)
Supergirl (2015)
Episode list - season 1
Starring: Calista Flockhart, Melissa Benoist, Mehcad Brooks, David Harewood
More mistakes in Supergirl
Cat Grant: All four of you standing there doing nothing, you look like the attractive yet non-threatening, racially diverse cast of a CW show.
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.
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.