I would recommend taking a simple arpeggio walking line and permutate it in the circle of 5ths. Like this 2-5-1-1 line in G, for example:
C-------------------l--------------------l--------------------l---------------------l
G-------------------l--------------------l--------------------l---------------------l
D---------------7---l--------------------l----------------5---l---------------------l
A-------3---7-------l----------------5---l-------2---5--------l---5---2--------2---l
E---5---------------l-------2---5--------l---3----------------l------------3--------l
B-------------------l---3-----------------l--------------------l--------------------l
then permutate it up a fifth into D:
C-------------------l--------------------l--------------------l---------------------l
G---------------9---l--------------------l----------------7---l---------------------l
D-------5---9-------l----------------7---l-------4---7--------l---7---4--------4---l
A---7---------------l-------4---7--------l---5----------------l------------5--------l
E-------------------l---5-----------------l--------------------l--------------------l
B-------------------l---------------------l--------------------l---------------------l
And then into A:
C---------------11--l--------------------l----------------9---l---------------------l
G-------7---11------l----------------9---l-------5---9--------l---9---6--------6---l
D---9---------------l-------6---9--------l---7----------------l------------7--------l
A-------------------l---7-----------------l--------------------l--------------------l
E-------------------l---------------------l--------------------l---------------------l
B-------------------l---------------------l--------------------l---------------------l
And then into E (up a fifth, down an octave):
C-------------------l--------------------l---------------------l--------------------l
G---------------11--l--------------------l----------------9---l---------------------l
D-------7---11------l----------------9---l-------5---9--------l---9---6--------6---l
A---9---------------l-------6---9--------l---7----------------l------------7--------l
E-------------------l---7-----------------l--------------------l--------------------l
B-------------------l---------------------l--------------------l---------------------l
And now B (again, up a fifth, down an octave):
C-------------------l--------------------l---------------------l--------------------l
G-------------------l--------------------l---------------------l--------------------l
D---------------11--l--------------------l----------------9---l---------------------l
A-------7---11------l----------------9---l-------5---9--------l---9---6--------6---l
E---9---------------l-------6---9--------l---7----------------l------------7--------l
B-------------------l---7-----------------l--------------------l--------------------l
Alright, after an hour of tabbing, I hope that helps you

I hope I didn't screw anything up.
continue playing around the circle of fifths (F#,D#,G#,Eb,Bb,F,C, and finally back to G) in the pattern I showed. Start to recognize notes on the B and C strings, and their relation to the notes you already know on the EADG strings. Using the full range of the bass will help you get use to it.
Also, just playing extended scales through 2+ octaves will help, too.
Don't let that beautiful bass go to waste!
I hope that helped. (It had better help!)
Now I'm going to go watch The OC
Thank you, I'll be here all week.......year....
~Alex