I use a deep-dish pan gripper and a cake decorating spatula with a narrow bendable blade. I first grab the pan with the gripper, then run the spatula blade around the pan to make sure the crust is fully released from the pan, then slide the spatula blade down the side of the crust and with a slight flip of the wrist I slide the spatula blade under the pizza to help guide it out of the pan onto a cooling rack.
That's pretty much the same thing I do, which I guess is a trick that was branded into me while I was a driver at Pizza Hut a couple decades ago. Since I generally take pictures of my uncut deep dish pizzas immediately after pulling them from the oven, I usually don't serve them in the pan. To remove pizzas from the pan, I start by using a red American Metalcraft pan gripper
(model number N9494) to hold the pan. I like this gripper a lot better than the more common chrome grippers
, because the chrome gripper always makes me feel like I might drop the pan.
Holding the pan with my left hand, tilting it slightly, I use a very flimsy metal spatula-like device to lift one side of the pizza out of the pan with my right hand, then use the spatula to guide the pizza onto my cutting peel as I pull the pan away with my left hand. (Actually, I think there is a picture somewhere of me doing this. Let's see if I can find it.)
I don't know if the picture helps at all, but here it is.
EDIT: Oops, I guess this picture is from before I had the red pan gripper. Pot holders work, too. Note: The black plastic thing on the left side of the picture is sitting there because I quickly learned that it is useless for this purpose. After trying to use it to remove the pizza, I looked for something that I'd actually be able to get under the pizza. So I guess that picture is the first time I used the flimsy metal piece. Also, that makes me think you're probably just using the wrong kind of tool to get under the pizza and lift it easily, because I simply could not do it with the black plastic thing.