Otherwise, the advice I've usually heard is to purchase tickets as far in advance as possible.
Several years ago, when Amtrak was not sophisticated about yield management, they usually allocated some sleeper inventory into the lowest bucket at inventory release, 11 months in advance.
That has not been the case for a long time now, so the "buy at 11 months" advice is badly outdated. They had already moved away that practice before COVID and since COVID they have gotten increasingly aggressive in their yield management practices. Now they typically allocate all inventory into the highest couple buckets at the 11 month inventory release.
These days, 4-6 months in advance seems to be the window where they review developing demand and may reallocate some inventory into lower buckets. Personally, I check prices for a planned trip early on, 10 or 11 months in advance to get a baseline that is usually in a higher bucket, then leave it alone or check it irregularly every once in a while. At about 6 months out I start checking regularly, no less often than weekly, for a bucket drop. If I see one, I grab it, since Amtrak seldom allocates more than a couple accommodations into their lowest
open bucket (which usually is not the lowest possible bucket).
These days, booking that far in advance usually pretty much guarantees higher buckets regardless of travel date. The only reason to book that far in advance is to secure a desired accommodation on a specific date. Many people here do that, then track prices and call to modify to a lower price when they spot a bucket drop. Personally, I do not do that because I feel like there is no guarantee Amtrak will always continue their unpublished policy of honoring price drops when a customer spots one. If they stop that practice, they certainly are not going to announce it and if they do you will be left holding the bag. Still, I think it is a valid strategy, I just do not do it myself.