It's not until much later, after defeating Reactron that she has a free moment to relax and go over everything that's happened. With that comes the realization of something she's been waiting for what seemed like ages. Her soulmate.
You look taller on TV was written over her heart as any Krypton soulmark. It's been there sometime during or after crash landing in the Phantom Zone. She doesn't remember having it before she was found by her cousin but anytime before that is nearly a blur of emotions. She didn't even know what to make of it until Alex discovered it and told her all about Earth soulmates and their soulmarks. While Krypton had them, they were much rarer and often only in Kryptonese.
It didn't make losing her family or her purpose on Earth hurt any less, however it was somewhat comforting to have a person somewhere out there waiting for her. And it becomes so normal to look at eventually that she barely pays attention to it. She knows to keep an ear for it. However for nearly twelve years, nothing. She should have thought of it before, but she's been so preoccupied about becoming Supergirl and dealing with everything for Cat and James, that she forgets the major difference between then and now - her actions as Supergirl has her on television. She should have made the connection before, but there was too much going on to worry about it.
Her soulmate? Maxwell Lord.
CEO and founder of Lord Technologies and definitely not a fan of hers if anything he's said on news counts. Hell he even falsely states that Superman saved him instead of her.
Kara wonders if he had realized it himself yet, or if he often gets asked Where is he?. She wants to go to him, but he's not a huge Supergirl fan to begin with and she's not sure how to ask him or hell just tell him. There had been numerous times they've been in the same room due to his ties with Cat Grant but not once did they speak to each other before - hell, she doubts if he even knows her name! Of course this had to happen when she was Supergirl.
If she does go to him, this means he could hold this over her head. Just because they were soulmates it doesn't mean everything was sunshine and rainbows now. She knows this, has seen how much work soulmates have to put forth to have a relationship. Growing up, the Danvers constantly spoke about being cautious with her friends and her future soulmate. But with being Supergirl now? He's at risk, if it's ever discovered she had a soulmate or who it was to begin with. It's probably bad enough that he's the CEO of Lord Technologies and one of the smartest men on the planet. He'll probably be targeted again, not just by Reactron, but by others as well. How was she supposed to keep him safe and keep her silence?
What is she supposed to do now?