There are some obviously bad approaches that you can take to left hand technique (including position, alignment, fingering choices...), but there are also several potential "good" approaches that you can take -- these will depend to some degree on your physiology (how big your hand is, how flexible, how long and how even in length your fingers are, how broad your palm is across the knuckles) and to a great degree on your implementation of them. If you have, as you say, established a solid floating-thumb technique with your right hand (no mean feat), then you are capable of practicing to establish a solid left hand technique.
I said "potential 'good'" rather than simply "good" approaches, above, because you can make a perfectly valid technique bad if you approach it in the wrong way. I'm not going to talk about specifics of technique, but I will suggest a few things to focus on as you practice whatever approach you decide on as best for you:
1) Balance 2) Efficiency 3) Static Comfort 4) Fluidity/Flow
1) Balance: As you fret a note, your left hand should feel balanced on that finger -- you should feel that you can easily establish an equal amount of weight on any one of the four fingers without your hand straining or wanting to collapse to one side or the other. At the same time...
2) Efficiency: It's possible to hold one note and have your hand balanced in a position that will make it impractical to move to another note with any speed. You don't want "flying fingers" or excessive rotation/flopping about, but you don't want to lock your hand in an uncomfortable position, either. In the video posted of Sean Hurley demoing the P-bass, he looks very comfortable and efficient in his movements. He is barely moving his fingers (someone mentioned Rocco Prestia). I don't know that that's practical for every style of play (I suspect that you'd see some differences if he was playing more melodic, sustained stuff), but what he's doing nicely shows a comfortable left-hand set without a lot of unnecessary movement. At the same time, if you study what he's doing closely, you'll note that he's not holding his left hand rigidly in one alignment or one shape -- there are subtle expansions and contractions of the hand and some rotation of the wrist as he goes for different things. This leads into 3) and 4)...
3) Static Comfort: You can practice something I've heard called "posing" with reference to bowed string technique. Try playing through something (not too fast) and stopping yourself -- as if someone shouted, "Freeze!" -- at various points. Check your alignment. Is this position comfortable, or do you want to get out of it as quickly as possible? If the latter, see what you can do to fix it. Then incorporate it back into the movement (rinse and repeat)...
4) Fluidity/Flow: Whatever you do with 1) and 3) has to make sense with 2) and 4). If you're practicing something that you're going to play fast, practice it slowly, but practice it with the technique that you'll need in order to make it happen quickly. If I'm playing a melodic line and land, with my 3rd finger, on a long G at the 12th fret, I might lift up all my other fingers and employ some vibrato. This makes sense in that context. It doesn't make sense in the context of breaking down a tricky run. In the latter case, I'll practice slowly, but I'll try to employ the same efficient, "low-to-the-ground" movements I'll employ when (I hope) I'm playing the run at full tempo. So far, this sounds a lot like 2). They are closely related and could perhaps have been combined into one point, but the emphasis here is on focusing on how each movement leads/flows into the next. You could say that the focus is on transitions (including through the "in-between" movements and positions), from one finger to the next, one chord to the next, one position to the next -- and making sure you iron out any wrinkles that you find (see point 3) and that everything flows as smoothly as you can make it.
Everything above relates to everything else, and none if it is intended to be dogma -- you could as easily condense it to one or two points or expand it to ten -- and you can certainly overthink it. I just thought about what I focus on in my own technical practice (when I'm disciplined enough to really practice as opposed to just play stuff) and this seemed a logical way to break it down into areas of focus.
A good teacher can help speed along the above process a great deal and can help you avoid potential minefields/missteps. Even with a good teacher, though, you have to practice and you have to develop some capacity for self-correction.
Emphasis on good teacher, by the way. There are plenty of bad doctors, and they all had to go through years of schooling and pass a presumably stringent certification process to do what they do. There is no such requirement to give bass lessons -- as always, pay attention and use your judgment.
Good luck, and have fun.
