It’s great that you have some room in your budget to make cuts. Work on cutting back unnecessary expenses, and look at ways of earning money on the side to help you get to your goal of 3 months of emergency savings, which for you is this: {calculation}
Once you reach this goal, buckle down and do it all over again, and you'll be at 6 months. Sometimes, chipping away at the overall goal is the best way to do it.