Atlanta appellate attorney Amy Weil says she hands out her business cards whenever opportunities arise. A number of those cards and a professional connection to Justice Clarence Thomas earned her a U.S. Supreme Court argument that many lawyers would covet.

Weil of The Weil Firm was picked last year to argue as a friend of the court, supporting an appeals court decision that the U.S. Justice Department had chosen no longer to defend. It was a rare opportunity—the court picks amicus counsel only a handful of times every year, and those slots have often gone to former Supreme Court clerks.