If you are teaching on an E-2 Visa, then unfortunately you are tied down to your employer until the end of the contract.
Even if you have signed your next contract with the new school, if you quit working for your current school now, then your E-2 visa will be cancelled by your current employer, and you must leave the country within 14 days.
There is a way around this. In order for you to "quit" your current job and stay in Korea by switching over to a temporary D-10 visa until your next school secures another E-2 visa, you would need to get a paper called "a letter of release," from your current school. Here is where the problem comes in, the employer is not obligated to provide you with this letter of release, and in 99.9% of cases will refuse to do so as it is not beneficial for them to lose an employee. If you try to bring this up with them, it might backfire on you and they could try to terminate you early, or make your last 3 months a living hell.
Since you have already found a nicer school to work for, I would honestly say, do you best to get through the next 3 months, collect all of your paychecks and contract completion bonuses, and transfer quietly.
Lastly, I hope you were able to talk to some of the teachers currently working at the new school that you'll be going to next March. While an interview can go well and the school owners can seem "nice" on camera, with Hakwons you never really know what you're getting yourself into until you really start working there. Since E-2 visa holders here are practically "slaves" to their Hakwons, as the school owns your visa for the duration of your contract, and Hakwons are private businesses out to make as much profit as possible, with the owners investing their life savings and hoping this "Hakwon" will keep them alive till the end of their days......please, please do thorough research before you commit to a life of misery for a year.
I don't want to be the Debbie downer here, but I want to give you something to keep in mind as it seems that you are new to working in Korea. The whole Hakwon industry is built on "Competition" and "ignorance" for a lack of a better word. In this competitive country, where natural resources are scarce, and the biggest resource being "humans" where everyone's life goal is to beat their neighbor, parents spend hundreds of thousands of dollars on educating their children in many subjects and especially "English," the language they themselves don't know much but hope their children can master for an opportunity to travel, study, work, or even immigrate to either the USA, Canada, the UK, Australia, or get into one of the top universities in Seoul and grab a high paying job at Samsung, Lg, ect... On the other side of the spectrum, in order for these Hakwons to remain sustainable, the owners rely on "naive" young Native English Teachers, who come here completely "ignorant" or unaware of the low salaries, their legal rights, the Korean language, the fact that they don't even own their own E-2 visa (like in Japan), but their visas are owned by their employers, to keep this system a float.
Before you sign your contract with the new school (if you haven't already), I would suggest asking if the Hakwon only hires E-2 visa teachers (slaves), or if they also hire Native English teachers on F visas. For example, (F-4 ) Overseas Koreans Visa, F-6 (Marriage immigrant visa), or (F-5) Permanent resident Visa. This is important because the F Visa holders own their own visas and they can quit their jobs or change jobs whenever they want without needing "a letter of release" and the Hakwon owners know this. So the bad ones who want to abuse their staff will try to say hiring 'F-visa holders is illegal" when in reality they won't do it because of the consequences.
I really wish you all the best at your new school! Right now, it is still possible to save quite a bit of money working in ESL in Korea if you're young single, and frugal. Save up as much as possible, and move on to bigger and better things in life!
