Note that the notes I whistled would complete the 2-octave fingering if I only had a 6 string bass. In that case I would end on my starting finger (2nd finger in that case), only 2 octaves higher and one fret lower (thanks to the little shift between 5th and 6th scale degree during your second octave). And considering that a 6 string bass has a B as your lowest, and a C as your highest string, you must end up one fret lower ;-).

Now, likewise, if you start on your 4th finger, you can continue with your 2nd finger version after one octave:


















 

Note that again, the notes I whistled would complete the 2 octave fingering and let you arrive on your starting note with your starting (i.e. 4th) finger, only 2 octaves higher and one fret lower. See a pattern / divine system here???
If you had a bass with an infinite number of strings (thank God you don't!!), you could continue playing that 2 octave fingering until you make your dog scream (past 16 kHz or so), moving one fret lower every 2 octaves (compliments of Mr. Pythagoras)...

So, how can I put the system to work? Here's C major all across the neck (excluding open strings), ascending:



















 

And descending:



















 

(Continued!)