If all you need is $10-$12 an hour, I would suggest you look at small firms in SC.
You can definitely work a law job over the summer and get paid. I had friends below the median at a T2 who worked for small firms and solos, making $10-$15 an hour in a cheap COL area. If you're allowed to use your Lexis/Westlaw account over the summer, you've got a very valuable commodity to a small firm or solo who doesn't have an unlimited search contract.
1) Plan a trip to South Carolina that will allow you to interview (spend 2-5 weekdays), either in January or spring break.
2) Go to findlaw.com, martindale.com, and if need be, yellowpages.com, and look up solo lawyers and small firms practicing in a field that interests you. Also use Martindale and/or ask your CSO to see if there are any alumni of your school in those cities.
3) Send them targeted cover letters with a resume (get your CSO's advice about whether to include your GPA), explaining your desire to practice in CITY and get more exposure to PRACTICE AREA. Also tell them you will be available for an interview in CITY on DATES (or whenever if you're close enough to make a short trip).
4) About 10 days after sending the letter, follow up with a phone call if you haven't heard anything.
It's probably too late to interview in January. Interviewing during spring break might be more viable time-wise, but it's a little late in the semester. If you're close enough to SC that you could drive and only miss 1-2 days of class, then you can do it piecemeal at any time.