Over this summer, participate in the legal job fairs aimed at students. The firms will hire a few students through these programs. You may be able to go through the process before they even start looking at students through OCI. If things don't work out with the firms you liked, you can still apply with them again during Fall OCI and get them to look at you again. You get two chances at them, and if they don't like you, they just tell you no twice.
That doesn't help with the resume, but does increase the chances of getting seen by the person who likes you. For resume, it'll be grades, law review, etc.