Moving is an unsteady business. Some weeks will be busy all week. And some weeks might be only one or two scheduled moves. Most people tend to move around summertime. This is called busy season. The beginning and end of the weeks and months also tend to be busier. The middle of both are generally more available.
Plan ahead of time if you have a specific date in mind. Slots fill up quickly. Winter months are more flexible. Not a lot of people move in the harsh winter months. Especially during the holiday season. Choose your mover and date wisely. Plan it all out from start to finish.