In my experience, yes. You should, at minimum, get the difference between coach and sleeper for the connection with the missed leg.

The one time I had a missed connection in Chicago (between two long distance trains, missed by a couple minutes! Long list of failures on that trip...) I was given a room on the next day's train, along with a taxi voucher, a $10 food voucher, and a room at the Swissotel for the night. After talking with the agent, she said we also had the option of going to the Greyhound counter and getting tickets on Greyhound if we wanted. We went that route as we desperately wanted to get back that day, and there was an itinerary that would do so. (There were multiple a day, and the agent was flexible enough to get us on the earlier one that left from the Greyhound station and would get us home earlier, even though the stops were slightly different than what the Amtrak-coded one normally allows.) Once we were on board the bus, we called AGR and we got a full refund for that segment of the trip, even though Amtrak did pay for the bus home from Chicago.