Rejected by most of sophomore pre-internship programs?

Reach out to recruiters and let them know you're interested in learning about the recruitment process at their firms. (if you're a diverse candidate focus on diversity recruiters) then ask them to connect you with analysts and associates within the group/division you want to work in, also reach out to MD's in all divisions-letting them know you want them as a mentor, etc... build a good network and follow up with your progress leading up to fall interviews and you'll hopefully have a good shot at a first round which would put you at equal standing with people that got pre internship event because at that point its all about how well you interview (so prepare well!)

Honestly, I don't think it matters as much as you think. If you put in the effort, you will be rewarded. Keep a high GPA and network, and you will have a good shot at getting a first round. After that, those sophomore programs don't really matter. It all comes down to how much the people at the bank like you over other people.

Assuming you are diversity/female then it's a function of your networking beforehand probably, or lack of previous finance experience/demonstrated interest in finance. If you're not diversity/female you won't get invites typically. Just keep networking for the SA process and you'll likely be fine.