There’s a lot of “wisdom” flying around when it comes to saving costs during a move, though there’s no clear answer for us:
Move during winter because demand is lower. That’s true, but winter moves can take longer due to snow and ice.
Move mid-week. Demand is usually higher on the weekend, but that doesn’t mean prices are. Your move is priced on the amount you are moving, not when.
Move in the morning. This makes sense because you’ll spend time packing and loading up during peak hour and drive when traffic is lighter.