I use my fan for this very reason. 'Fortunately' mine makes enough noise that simply turning it up to high works well enough so I use the fan speed and location within the house to fine tune things to my needs.
White noise from the fan is caused by the sound of air movement, especially turbulent air flow as it moves around items and itself in a pattern much like river rapids. What will tend to cause more of that is inefficiencies in the air flow. The primary items interacting with the air flow is the blades of the fan. Therefore to create more white noise, I suggest roughing up the surface of the blades so that it creates more turbulence.
The front edge of the blades is what hits the air first while the face of the blade is in contact with the air much longer. I would expect that roughing up the front edge would tend to create more loudness while roughing up the blade face would create more white-ish noise.
This could be done permanently by marking up the blades with something like sandpaper, or a grinder, or a razor knife. To do temporary, which also allows for more adjustment and experimentation, you can add tape, or even better, hot glue melt pieces on in a 'bumpy' pattern on the surface of the blades.
Note that this will be adding weight to the blades and we want to add weight to each blade evenly to keep the overall rotating fan in balance. When out of balance it will wobble and and likely make undesirable non-white noise, and tend to wear out the fan much faster due to stress on the bearings.