HVAC flanking path noise transmission is my arch enemy. I'm guessing this closet doesn't have HVAC ductwork, but even if it runs nearby....

Mac is right that eggcrates break up reflections and disperse them. Like a nice book case with staggered depth books does the same.

Wood floor, I would at least have a temporary way to cover that up. In fact, I would carpet it, and if you do want a more reflective floor, then throw down a piece of paneling.