Deferred, Can I Improve my SAT Score?

Hey everyone! I was recently deferred from my top choice school, and was looking for ways to stand out in the regular decision pool. I was thinking of retaking the SAT but want to decide if it’s worth it. I’ve taken it once before, and scored a 2020 with very little prep. So my question is twofold. First, what would be a significant enough score improvement to justify notifying this school with the update, and second, could I realistically achieve that level of improvement with a little more than a month to prep?