It all depends on the size of your move and its destination. if your destination is from coast to coast then it will require planning a head of time such as 5-6 weeks in advance.
 

If you moving within your city limits 2-3 weeks is quite enough for planning and scheduling with the movers.
 

For example moving from a big house its best to get at least 5-6 estimates to get the best rate for your move, we would recommend to start planning 5-6 weeks in advance.
 

At Summer time its always better to set time in advance as most of the movers are extremely busy during the summer. do not wait for the last minute.
 

In any case its not recommended waiting for the last minute, at summer times the end of the month is most booked, it will be better not to move at the last day, it might cause delays or it will be more expensive. plan ahead in order to save time and avoid unnecessary delays.