EHX Bass Soul Food is an excellent budget pedal, so I'd recommend trying one of those. If you've already got an OD unit and want to keep the low end then I'd look at a blend pedal that will allow you to mix clean, un-overdriven signal with a dirty, gnarly effect, thereby maintaining your low end.
Be aware though, that the studio recording you're referencing will more than likely be recorded this way (take a clean feed directly from the bass to mix with the overdriven signal) which makes replicating the sound somewhat tricky. My own personal favourite is the Dave Hall VT1 DI-EQ, but I understand that it's not a budget pedal - add a dash of the Waves R Bass plugin in the studio and it's every bass sound you'll ever need. Hard to replicate live though.