When I shut my car off, the power plug shuts off as well. I know in some vehicles, there is a fuse placement change that can make these sockets on all the time. I have a 2013 GS 350. I'm wondering if I have such an option for this car?

 

Some rewiring (and tapping) would be needed to make it hot all the time. Why do you want it to be hot continuously? Having a continuous load will draw down the battery...

 

Here's a workaround that will protect your car's battery. Buy a power bank with pass-through charging capability. You can even get one with a cigarette lighter outlet
This way your device can continue to draw power while the car is turned off.

There are also plenty of less expensive options if you just need a USB port for charging.