Nic, it isn't the company that hires a driver, in expediting, (though they and their insurance have standards you must fulfill), it's the owners of the trucks, whether they have just one truck, or a fleet.
Read the Newbies, & Newbies FAQ fora, and check out the ads posted here, to get an idea of what the owners are looking for.
One bit of advice, if you don't mind: if and when you begin calling to inquire about driving for an owner, be prepared, with a short list of specific questions, so as not to waste their time, and your own.