Short of expanding their space, there's a way to make said space hold twice as many CDs. The solution is as simple as swapping out the CD trays with ones that can hold 2 CDs.
Beneath the box in the image, there's a 4-drawer CD Chest with 384-CDs-in-single-cases capacity. It's larger then said box. My CD collection has clearly outgrown it. Until now.
Step 1: What you need.
You can buy these at SleeveCity.
I ordered a 100-Pack of 2-CD Smart Tray All Clear for $35.30 USD.
$28.00 + $7.30 for least expensive shipping option.
This covered about 1/3 of my collection. I'll get a 300-pack later and do the rest, and post the results.