You have to be prepared for a situation like this. If you will be leaving the pick up location, arrange for someone else to store the vehicle if a carrier is not found in time. If you need the car delivered by a certain date be prepared for it not working out. And if your auto transport budget is severely limited be ready to wait a long time for a carrier. If your pick up or delivery date has to be exact try to make the other as flexible as possible, for example, if you need the car delivered between the 22nd and 23rd make the car available two weeks ahead of time (or more depending on the distance the car has to travel).
Be proactive. Make your car easier to pick up and drop off by offering to meet up in larger cities and increase flexibility in dates where possible. Your best move, however, is to raise the price. This is the most likely reason a car is not getting picked up; specific dates and low prices do not mix well.
Patience is the key. If you’ve chosen a competent company the only hold up is that no available trucks are in the area. Don’t start calling every broker you can, this just creates confusion and could scare away potential drivers. Contacting multiple brokers is good to get an idea of the price but choose one to actually move your car.