Unless you have status with a rental car company forget the free upgrade. (it's possible -- just highly unlikely)
I have status and I am occasionally charged for a convertible upgrade. Aside from free upgrades, I've paid as little as $8 per day in San Diego and up to $25 per day in LA. In LA it would have been cheaper to outright reserve the convertible in the first place. Oh well.
Now if you can get access to the Emerald Aisle with National Car there is a chance there will be a convertible on the lot which you can grab for free. I guess you could hang around for awhile and see if they bring one in too.