Carol wouldn't have necessarily understood Clark's hesitance to share his name. For her, it was the opposite. She held her name closely and wore it proudly because it had been stolen from her for so many years. Vers had been a lie. Carol was who she really was, and even though some people called her Marvel (which she still never really understood), she knew it was just easier for people to remember. And probably easier for the military to explain since she was technically still "dead" in their official records.
She glanced back at the tree and sighed. DId she admit it? "I'm always looking, I guess?" She looked back towards him and nodded. "Why? You interested?" That was definitely an intriguing offer.
True, Carol didn't really know this Kal...but he was wearing a cape and he could fly, so she was assuming he might have an idea about what he was doing. Maybe.