Are the schools are your list test optional this year? If not, I would suggest that you take the SSAT or ISEE. You will also need to take TOEFL or Duolingo, unless you are currently attending an international school where 100% of courses are taught in English.
If you are ethnically Korean, all the schools on your list are going to expect you to score in the 90s on the SSAT. This is non-negotiable, unless you are a national-level athlete or have another strong hook. Among those, SPS historically has had a larger contingent of Koreans, due to the history of chaebol/diplomat kids since the 80s. However, if swimming is your passion, then it might not be worthwhile pursuing. Lawrenceville and the Phillips schools might serve you well in that regard, but the bigger class size also means that you will face stiffer competition during tryout or early evaluation.
It is a great sign that you are in dialogue with the swim coach at Hill. Try and do the same with other schools, and if you can get endorsement from the coaches, it will greatly increase your chances of admission.
The general (and very sensible) advice here is to cast a wider net. Applying to 8 top schools does not mean that your chances will be improved by 8x. Also, Cate is a particularly small place, so the spots will be very few and you should adjust your expectations accordingly.
Good luck!