I'm a bit late to this thread i appreciate, but the strymon Deco adds all the saturation and overtones i could ask for, into my fairly clinical genz benz head (which i love like my only child). Its effectively a tape recorder saturation and flutter/wow emulation pedal.
here's a great Italian bass demo, the satutation solo`d starts at 3:10. https://www.youtube.com/watch?v=4c3SQjePF5I
The saturation side of the pedal is by no means an obvious "effect" but gives a subtle warmth to the sound that wont change the core tone like all the pedals already mentioned on this thread.
The flutter side of the pedal can do everything from subtle tape warble, lush chorus and into double tracking. And it has stereo outs if your into that sort of thing!