Greyhound...Cross country it is not the way to go....If Greyhound is the best the USA provides for transcontinental bus service, it speaks poorly for the country.
AFAIK, Greyhound is the only ~transcontinental~ bus service in the USA. But seriously, if one is going all the way between the oceans, put a pencil to paper and figure how much will be spent on several days' food for bus (or rail) travel, and take another look for cheap air flights.
BTW, air, train and intercity bus travel modes: guess which of these got no zilch nada dedicated pandemic disruption assistance money from the American government? Now guess which one serves the most American communities? Aaaand finally guess which one creates the least greenhouse gas per passenger mile? Hint: it's the one that Adventure Cycling has no partnerships or programs with and doesn't promote.
But anyhoo Greyhound is not as good as bus service gets in the USA. (That would be Vonlane.
) Things are still unsettled because of the pandemic (it's not like the government helped bus companies out, remember) but as things vector to normal, here's an idea. If your bicycle adventures are in the Seattle-San Diego-Phoenix-Salt Lake tetragon or in the Boston-Miami-San Antonio triangle, check out FlixBus. They're a fairly recent entrant to the US intercity bus scene from Germany. Yeah, they're experimenting with battery electric and hydrogen fuel cell buses to become greener in the future, but the exciting thing for cycle tourists right now is many of their buses have bike racks. Yeah, actual bike racks. They'll haul your bike and you don't even have to box it. You reserve a spot for your bike when you buy your ticket. Unlike Amtrak, you can put your bike on or get it off anywhere they stop. If demand is there, FlixBus says in the future they could haul bikes on a rack plus up to 5 unboxed bikes in the luggage hold.
Another option. Options are good. Check it out. It might work for you.