An overdrive is a really kind of mild distortion.
Just a small amount of breakup and dirt like a pushed amp.
My favorite is the Holy Fire by Creation Audio. They are coming out with a bass version of this same pedal. It's not out yet but the (guitar version) holy fire is amazing and I've gone through a good dozen of overdrive. It's expensive and it comes with it own power supply but what a tone!
Other pedal to consider that I have not tried: the new MXR bass overdrive, the Pork Loin which is capable of overdrive to light fuzz with everything in between and the Zvex Basstortion.
Just a small amount of breakup and dirt like a pushed amp.
My favorite is the Holy Fire by Creation Audio. They are coming out with a bass version of this same pedal. It's not out yet but the (guitar version) holy fire is amazing and I've gone through a good dozen of overdrive. It's expensive and it comes with it own power supply but what a tone!
Other pedal to consider that I have not tried: the new MXR bass overdrive, the Pork Loin which is capable of overdrive to light fuzz with everything in between and the Zvex Basstortion.