I've done this multiple times. Give them a call once the rental has started and they'll be able to tell you the new price without actually making the change, then if you want to go ahead they'll change it. Technically they can do it at drop-off too, but that'll be much more effort as you'll need to go to the counter to get the change made before they check-in the vehicle.

In my experience, the price has always been exactly the same as what it would have been if I'd changed it before pickup. If you go into your reservation and start the change process - keeping the box about using your original rate ticket - you'll be able to see that price. I don't know for a fact it'll always be that price, but it always has been for me.