The Neutral start was a weird one, my '57 Royal Lancer and my '57 Custom Royal Lancer both have it. It was only used by Dodge, only with a 3 speed Torqueflite. The 2 speed Powerflite didn't get it, it used the key. Plymouth, Chrysler, and DeSoto all used the key, no matter 3 speed or 2 speed.
OK, my 59 Dodge was originally a Powerflite (62 Torqueflite now, operated by 59 Torqueflite pushbuttons). Maybe the Torqueflite equipped 59 Dodges had the neutral button start. I never thought to check the 59 Torqueflite pushbutton assembly I got off ebay for a start button hookup. Maybe it's there. ??