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While most law school application deadlines have come and gone by the time March rolls around, deadlines don't always matter. In fact, a quick web search will turn up countless tales of non-waitlisted law students being admitted even as late as July or August.
However, there is definitely a catch when it comes to applying late (which is generally considered any time after January). If you don't have some sort of wow factor that the school actually needs, or some alternative way in, you might just be wasting your time. Additionally, the conventional wisdom of applying early being better than applying late is absolutely true; schools just have more open seats in the fall than in the spring, where usually they are sorting out acceptances and the waitlist.
Skewing the Numbers Up
If you took the LSAT late but managed to earn a score that causes jaws to drop, and you have a high GPA to match, you might be able to write your own ticket at any time. Most admissions offices are hyper concerned with bringing up their average numbers for the class. And as one law school admissions consultant notes, if you're going to be applying late, you better have something to help the school's admissions office.
Foot in the Door
If you have some way to get your foot in the door with the admissions folks, then submitting an application late isn't the worst idea. And if you don't have a way in, you can try to make one yourself via networking, or even cold calling the right people. However, if you're going to try to create an in where none existed, be prepared accept rejection with poise and as politely as possible, as you may want to apply again for the following term.
Is It Worth the Money?
The law school application process is not cheap, and if you aren't significantly better looking on paper, or a celebrity (or celebrity's child), spending the money to submit a bunch of late applications probably isn't the most fiscally responsible move. Alternatively, rather than applying late, consider applying for the next class as early as possible. There are countless volunteer and job opportunities out there to fill that gap year which will actually benefit you on a law school application.