Won't work. Although it is sometimes referred to as TB>HDMI, it is actually MDP>HDMI. So when you plug the adapter in, which looks just like a TB adapter, the COMPUTER recognizes it as a MDP adapter, and sends a DP signal out. The Thunderbolt Display does not pass through DP to it's TB port.
Oddly enough, if you plugged in most run-of-the-mill TB devices...like, say an external TB hard drive that has two TB ports on it, you can plug one TB port into the computer and the other TB port into a MDP>HDMI adapter and the TB external enclosure will pass the DP protocol between the computer and display. However, Apple has decided not to do this with their Thunderbolt monitor. It will daisy chain to another ATD.
Now, you can try this if you have all of the stuff already (it has been rumored to work, IDK for sure). Plug your Thunderbolt monitor into your computer. Then plug a TB external enclosure into the extra port on the thunderbolt monitor. Now, plug your MDP>HDMI adapter into the second port on that TB enclosure and it may pass the signal on.
Again, I don't know if this works, I've just read others have tried it. I still think the ATD may be the end of the line for the DP protocol....but if you already have the equipment, it won't hurt to try!
Otherwise, you are stuck with the HDMI out as the second monitor