I don't think you are risking your job. If they don't approve your expense, you owe the company the money for the ticket. I have had to buy a ticket from my house to a place where I could catch a jumpseat. It didn't fit the x and y thing.

You can stop enroute for 24 hours too. So you could buy a ticket from your house to the personal destination, stop for 24 hours and then buy a ticket into Memphis. Or jumpseat into Memphis to complete that trip.