As others have already begun to point out in the days following the amendment's rejection, the legislation was essentially toothless from the start. The federal government does not, after all, have the constitutional authority to prohibit states from requiring GMO labeling, let alone permit it. The federal government has also never even tried, at this point, to legally stop individual states from mandating GMO labeling. This means S. Amdt. 2310 was a faulty attempt to address an issue that is not even an issue, and one that attempted to do so using an unconstitutional approach.
A key thing to remember in all this is that neither the federal government nor the U.S. Congress has any constitutional authority to grant states permission to label or not to label GMOs. Under the U.S. Constitution, individual states already possess their own inherent authority to determine how they wish to handle the GMO labeling issue, and the federal government does not legally possess any authority whatsoever in the matter.