Don't wait for the last minute either. Give it at least 2 week before the move when you contact movers. Most interstate moves are done on large tractor trailers that typically hold 5 or 6 or more different household goods shipments. Movers rates are based on moving fully loaded trailers to and from origin to destination. That is why they need flexibility on the dates often for both loading and delivery.
Movers tend to get busier at the end of the month.
The last week of the month in May, June, July and August are the busiest weeks of the year. If you must move during these weeks, contact your mover well in advance and reserve your move day. It's not unusual for these weeks to book up 4 to 6 weeks in advance. If you can avoid these busiest weeks, you will be better off. All movers hire temporary, less qualified help to augment their work force during these peak times.