Many major moving companies in the US offer last-minute moving packages, even for long-distance moves. For a fee, you can often choose priority moving services that focus on getting you on the schedule, packing up your home, and driving to your new space as quickly as possible.
You may also want to consider storing a portion of your items and only moving essentials on the first trip. This option allows you to sort through your household goods later and determine what to move and what to donate.