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You can also get some teaching experience under your belt as a law student by working as a teaching assistant for undergraduate courses or for some law school courses.
If the teaching opportunities open to you will give you a chance to demonstrate your teaching skills and be evaluated on them by students and you do a bang-up job, that will be a mild plus with elite schools, and can help to counteract concerns about teaching that are generated by a less than stellar job talk. But teaching experience will not count for a lot, as compared to your scholarly and academic record. At less elite schools, in contrast, a successful teaching record may well count for more, particularly if it demonstrates your competence in core areas of the curriculum.
Even if a teaching assistantship does not help you gain credibility with a hiring committee, you may find it useful in figuring out whether you like teaching and whether you are good at it. In addition, your first experience teaching in an academic position will be less daunting if you do have some prior teaching experience.