| First off, I can't hear the sound on that video since I'm at work. As such, maybe the thunk sound you're looking for is OD or an octave. I have no idea.
That said, I think you're 'sort of' in the right place. There may be effects that can help you with tone, such as an exciter or a preamp (or maybe the aforementioned overdrive), but most of what constitutes good 'tone' on a clean-sounding bass can't be created by a pedal. It has to be in your signal chain elsewhere.
My experience is that good tone is not a percentage of the quality of the bass or how good your amp is, but more of like a chain. Your tone's sort of only as good (more or less) as the weakest point. My experience is that four main points that constitutes tone are your hands/technique, the quality of your instrument, the quality of the amp and the listening environment (bar room, venue, headphones, etc.). A killer player can usually pull a better tone out of a cheap instrument and a crappy amp then a lousy player with pretty good equipment.
I've also found that the weakest point is usually technique. Unfortunately, you can't buy technique, and life usually gets in the way, too.
As such, if you're looking for "tone" specifically, you may be able to find a little better tone with a good tube preamp, a harmonic exciter. However, you're probably better off getting better technique, basses or amps in that order.
Lastly, the biggest destroyer of tone is often a crappy sounding room where certain frequencies get bounced around more then others. This is more common then not when you're playing live, and the further you get from your amp, the worse the tone gets as more surfaces reflect or absorb certain frequencies but not others affect the listener. As such, what sounds great on stage often can sound muddy and befuddled in most of the club. It's important to remember to get a tone that's good for the room, not just you on stage.
It's also worse when you play loudly with a loud band in a live situation. Guitars or other instruments that are too loud create competing/similar frequency ranges that get accentuated with volume. The louder things are, the worse things usually sound. As such, if you're playing in a band with a loud guitarist with the bass turned up on his amp or a keyboardist with an overactive left hand, your tone will suffer, too. | 
