Summer months are the busiest months of the year for the moving industry. Moving during the summer months will be more difficult to schedule, as movers get booked up fast and early.
Beware that due to the "supply and demand", movers may raise prices, causing it to be more expensive to move during the summer. The same goes for the end of the month; this is when usually when more people are moving out and into new locations since leases often begin on the first of the month. It may cost you more and there could be unexpected delays.