Well one thing to realize is that getting a 163-165 is not easy it puts you in the top 13% of test takers. http://www.cambridgelsat.com/resources/data/lsat-percentiles-table/
. People that take the LSAT are college graduates who want to go to law school so they are generally quite bright and scoring n the top 13% is not easy.
This same logic will apply wherever you attend law school everybody is pretty confident they will be in the top 10% of their class, but again people that go to law school are good at school and 100% cannot be in the top 10%.
As to your score you stated you typically got a 160 in practice and a 156 overall that seems about right as many people do a little worse under the "REAL TIME and REAL STRESS" situations. However, one positive is that I believe most schools no longer average LSAT scores so if you retake you have nothing to lose and you also have plenty of time, but getting a 163-165 may not happen. If everybody was capable of getting into Harvard, Yale, UCLA, etc people would do it, but everybody has a ceiling.
156 can get you into plenty of ABA schools with a scholarship. I had a 3.2/158 and got into numerous schools with substantial scholarships.
I have also posted on this board several times about factors to consider when choosing law schools and going to the "best" one is not always best for your specific situation. If you want to pay a private tutor you can, but there are no guarantees and if you realistically did everything you could do it probably won't improve that much and in all honesty a 156 is not a terrible score you were in the 34% of people who are ambitious enough to go to law school and actually followed through with the test.