Display-connected hard drive not being recognised, requires replugging

I have an 11-inch MacBook Air from mid 2012 and a Thunderbolt Display. Every morning when I come into the office, I place the MacBook Air on my desk and hook up power and plug in the Thunderbolt cable. The computer wakes up, recognises the display and uses it, recognises the wired Ethernet connection plugged into the back of the screen and uses it, recognises the USB-connected headphone amp and uses it, which is all great.

What it does not do is recognise the USB-drive which is connected to a USB port on the back of the display.

• If I then unplug the Thunderbolt cable and plug it back in, it does recognise and mount that drive.
• I've tried different drives, and the problem persists.

I'm confused as to what could be causing this behavior and if there is anything I can do to fix this.