Hello again. Thanks for the recommendations. I've really come to the conclusion that every bass itself provides its own lows and weight in its tone. If my bass doesn't have the lows I'm looking for, neither changing the pickups nor the strings will give it to me.
In my experience, Precision basses with thicker necks have always given me a deeper tone, while those with thinner necks have always given me less body. This one has a thin neck, so I'm expecting something that it can't give it to me. I've tried changing the pickups, but the bass still sounds very similar, like itself. In this case I guess there's nothing I can do to make it give me something it doesn't naturally possess. The EQ in the amp can be my best option.
I usually use Elixir strings for their durability, but I find them very bright, and they maintain that brightness for a long time. Perhaps "normal" strings would lose their brightness over time and give me a rounder, more vintage sound. I've felt that way sometimes. All new strings offer too much brightness for my taste, so probably any set of old strings would work. Thanks again.