Milkman
If you look at Tom McD's link, the front surface where the light rays exit the rain drop is also reflective like a "half-silvered mirror" so a portion is reflected back into the rain drop and further dispersed. A "second order" rainbow results when this portion of the light is reflected back and out. Multiple orders exist but are so dim as not to be seen. I have seen 1 or 2 photos that included third order rainbows and the color order is same as the first.
Vernon