You should consider any major that will encourage these skills. That concern is just one more reason to select a major that you find interesting and challenging. Your answer to that question will indicate what your major should be. Keep in mind that people tend to get better grades in majors that they find interesting. There are several sections to the LSAT that cover such areas as logic and reasoning skills.
Several companies offer LSAT preparatory classes that some find beneficial.
Contact some area law schools and ask them what they are looking for in a student. Interships are avaliable. More than one internship, in the same or different agencies, is possible. Allows students to review your draft paper with comments based on how the instructor grades them.