To give you some super cheap options for great sounding drives, I recommend you to try out some of Joyo's drives and amp emulations intended for guitar.
Oddly enough they doesn't suffer from any noticeable low end loss, if any, unlike their drive specifically made for bass, Moon Base, which I would stay away from.
Other than that I am amazed by the quality of these dead cheap pedals.
What I would recommend you in specific for what you are asking for, was trying out either the Joyo Orange Juice or Golden Face in some their higher drive settings, the Orange Juice supposedly emulating an Orange amp and the Golden Face supposedly a Marshall.
They both seem to be capable of some quite raw medium level distortion, the Golden Face seemingly a bit smoother than the gnarly grindy character of the Orange Juice (I own 2 Orange Juice pedals myself, both in use).
Alternatively to the Golden Face they also make the Joyo Bristish Sound pedal, which also is a Marshall amp emulator, but with more tone shaping options.
For a heavier hard hitting kind of distortion, I can warmly recommend the Joyo California Sound, which is supposedly a Mesa Boogie amp emulator.
I use it myself, and it works well even without clean blend, though be aware that it isn't really a gnarly or grindy type of distortion but has more of a smooth character, even if quite heavy with the drive button maxed, as I have.
Finally there's the Tech 21 GT2, intended as a guitar preamp/DI, of which Behringer make a super cheap but quite accurate clone, the Behringer GDI21.
(The Joyo pedals mentioned are by the way also clones of the Tech 21's character pedals)
Here's a demo from YouTube demonstrating the Tech 21 GT2's capabilities as bass overdrive/distortion:
I know for a fact Justin Chancellor (bass player from Tool), use the GT2 as part of his dirt section.