Amtrak offers discount tickets if you buy at least 30 days before you travel, and the ticket is good for a period of time. They divide the country into (I think) 3 zones, and you get a specific numb. of stopovers per zone, and you have to complete the whole trip within a certain time (45 days?) of your start date.

At least, a coupla times, when I was looking for a way to go from here to somewhere else for various reasons, this seems to be my memory of Amtrak's spiel.   Tw