If you are looking for an overdrive pedal that is able to emulate a slightly overdriven amplifier, you ought to choose a stompbox without IC chips.
Because of their frequency range the clipping of ICs is always hard.
And this hard-clipping is definitively not the sound, you are wanting if you are looking for a soft low gain sound.
Why do IC-based overdrive pedals sound that way?
Overdrive and booster pedals amplify the signal of the pickups.
First, I would like to illustrate how the signal processing in an extremely simplified way:
The guitar signal has nearly 0.3.
But depending on the string attack the signal peaks sometimes reach an amperage of 1.2V.
The fact, that inside of the pedal, this signal will be again amplified by a factor of 10 before it reaches the IC, the peaks can easily reach 12V.
If the overdrive pedal runs on a 9V battery or a 9V power supply, the IC is able to deal with signals up to approximately 9V.
Anything above will be clipped what leads to compression and distortion.
Although the subsequently fading signal will not sound overdriven, it will be louder than before.
The IC-clipping is more harsh and abrupt than the clipping of JFETs.
The transition between the signal without and with distortion varies from the transition that results from an overdriven tube.
By reducing the amplification before the IC via the volume pot, the signal certainly becomes cleaner, but this also decreases the input impedance. The result is the same as if you wind down the volume pot of the guitar: the signal becomes dull.
18V power supply or voltage doubler?
If a pedal runs on a 18V power supply or a voltage doubler (like for example the Klon Centaur) you can avoid this kind of distortion.
In this case, the clipping diodes will create the actual distortion and, eventually use also high pass/low pass filters.
But beware: An internal voltage doubler might cause besides noise (the Klon Centaur for example does not like the combination with several digital delays – even if each of them gets its own isolated power supply).
For all pedals without an internal voltage doubler the use of a 18V power supply can even work wonders.