I think what will happen is:

• incoming light sees a smaller aperture
• then that will diverge a bit when hitting the lens
• then that narrower beam hits the internal aperture
• then that arrives on the sensor.

What you will have is:

• an improved aperture (but not that good as if you did it with the aperture blades)
• much narrower useable image (wasted sensor area)
• vignetting, but not with a distict contour (because of the Fourier transform and the out-of-focus qualities, you will see a soft ringing on the borders most probably).
• a kind of lens softness/halo.(The closer you put the small hole to the lens, the less this is.)

In general, the amount of light entering the system will be stopped down by the small hole, so you need a much larger exposure time.

Also note that any imperfections of the small hole will be visible on the image, especially around small light sources as flare.

You will have a better DOF on a smaller useable area of the sensor, and around it you will have this brighter/darker softness or ringing and then almost blackness.

Note also that the small hole will may not only cause brightness but color ringing as well.

Anyway, this is a great fun to play with. If you make some photos, just for the edification of all of us, please post some results :-).