$1/mile isn't really that far off, considering IRS rules on auto expense reimbursement (fuel, maintenance, etc.) is $0.575/mile for 2015.
Considering an overall average speed will probably be something like 45mph, the driver is getting $19/hour, not including expenses - I'm $#@!uming they won't drive straight through and would need food and lodging along the way. So, 12 hours a day @ $19/hour = $229.50, minus $100 for hotel, minus $30 for meals, leaves $100/day for the driver, which isn't much.
Best bet (in my mind) is to go get settled, then take a week off and come back and do the drive yourself. You could make an adventure out of it!