A series of elements I'd be keen on:
- Show Krypton's destruction -- a relentless, intense evacuation -- from her POV. Her dad saves Kandor, but Brainiac, who had been passing himself as a goodie, betrays the bunch and "steals" the city. Her parents die, she makes it out. Her mission statement for the rest of the film is find Brainiac, free her people.
- On Earth, she doesn't know the language, customs, anything. And it'd be fun to establish that the Kryptonian set of values doesn't always align w Earth's, so it'd be a learning curve. You get to make her different from Superman in that regard. Make the cultural divide a key theme, bc it'd make for great and distinctive material. No Danvers parents, bc they'd simply be Kents Light. Make her an orphan, like in the Silver Age comics. Maybe create a supporting cast out of that.
- Make Superman a non-present presence, but beyond an "he's simply off screen" basis. Say there was a recent crisis that he averted prior to her arrival, and he disappeared as a result. Sucked into the Phantom Zone, maybe. Spiderverse's Parker was used entirely in the service of Miles Morales's story, so do the same here. In making a Supergirl solo, there's no way around acknowledging Superman while giving him a lesser role, as it's not his movie. The options are to either be ok with it or be ok with it. There's no Superman movie franchise to accommodate anymore. Use that.