GE Fan Controller with switched outlet?

I have a wall outlet that is mounted high on the wall that is currently controlled by a switch (not a smart switch). The purpose of the outlet is to power a mounted oscillating fan. This is not a ceiling fan, but something similar to this (click here).

My question is, can I replace the switch with a GE Fan Controller (GE 12730) and effectively have high/med/low/off control of the fan plugged into the outlet? Or is that a bad combination electrically? Thanks in advance.

I would say it’s a bad idea unless you know what kind of a speed controller is inside the fan.

Thanks. Is there something specific I could look for in an oscillating fan that would be compatible with this type of setup?

Depending on what kind of wattage is your fan. You can have a smart on/off plug or switch to turn on/off power to the fan but not fan speed.