Nebraska had an accidental experiment by omitting an age limit on their Safe Haven law. Parents drove thousands of miles to abandon their older children until the loophole was closed.
Good article. The most relevant quote: "Mr. Landry, the state director of children and family services, said parents and guardians were mistaken if they thought they could walk away from their responsibilities. The courts might even order them to pay child support." Furthermore, Nebraska has since changed the law, as you already noted; they permit only babies up to 30 days old.
Nebraska had an accidental experiment by omitting an age limit on their Safe Haven law. Parents drove thousands of miles to abandon their older children until the loophole was closed.
https://www.nytimes.com/2008/10/03/us/03omaha.html