I have the same problem. But looking at the schematic of the module, there is a boost converter that converts the 3.3V from the onboard regulator to a voltage of around 14V. This one is the cause of the noise on the power rail. The only countermeasure I could do is use a large capacitor (330uF) across the power input of the OLED. I also added a cap on position C7 which is currently unpopulated. The cap here is across the output of the onboard 3.3V regulator.
I am running of 5V by the way, but my main micro runs on 3.3 with its own regulator.