This is what I would do if I were in your situation.
Contact your BGO directly and find out exactly why you were not accepted. Was it because you lacked leadership? Interview went bad?, etc. He should be able to give you atleast some insight. Now for sport, lets say you lacked leadership, get involved with some student organization at Elon.
Sometimes the BGO will come back and say you were an excellent candidate, but the problem was there were 10 qualified people for the spot and they just couldn't offer you a spot. In this case strengthen the GPA and CFA, all you really can do.
Whether you like it or not, the Naval Academy is an engineering school. From what I have seen, it seems that the people who get accepted the second time around are the people who take plebe equivalent courses. Not taking physics or calc may be a negative weight on your application. If you really want Navy, I would strongly recommend taking physics and calc at the least. If you were on the admissions board and you had two second timers, one took plebe equivalent courses and the other took all humanities/liberal arts based courses, which candidate would you choose knowing the academy is an engineering school?
Double majoring and double minoring is probably a bit overkill, but hey thats up to you. Doubt if having a double minor, let a lone a minor period will do anything more for you post-college combined with a double major. Have you looked into the requirements for these majors/minors? A lot of times even liberal arts based majors require a math and lab science. In which case you would have to take them anyway, so you might as well do it to get a shot at the academy. I know that a polysci major at the public universities in GA require one lab based science and 6 credit hours of math. Other schools for example don't even let you really get into your major coursework until soph year.