If you specifically want the Stranglers sound get yourself a Tech 21 English Muffy - it's a two channel version of the original Leeds pedal, both designed to emulate a Hiwatt stack.
I've used the Leeds as my main preamp for years and with a Precision it gets very close to the Stranglers sound.
In my experience a dedicated overdrive or distortion pedal is always going to be too much - you need something designed to replicate a pushed amp. Run it in the effects return of your amp to bypass the preamp stage.
Best of all, Thomann have these Tech 21 pedals on clearance for £133!
He's probably using more top and drive than you'd want for JJ but it gives you an idea.