I know that these things are an eyesore, but I stumbled onto an easy fix to make these things blend in bit.
I'm fortunate enough to work across the street from a beach by a river; so during one of my usual riverside lunch breaks I found a twig (probably cottonwood) that had rotted on the inside, leaving perfectly intact bark.
Didn't take much to clean out the powdery wood inside:
After that, it was easy to just cut it lengthwise with a pair of scissors and wrap it around the filler tube:
I even made it higher than the actual tube, to hide the neon yellow earplug I use for a cap:
The next time I redo the false bottom - likely after I move in a couple of months - I'm going to make this a tad more permanent with hot glue.