So, reluctantly I'll add my two cents. First, a disclaimer, I'm not a professional bassist and don't pretend to be a great jazz bass improvisor, meaning, I'm a hair's breadth more advanced than you are - but then, that may be more helpful than Ray Brown's suggestions... you'll have to decide.
Here's what's worked for me and where I plan to go, your mileage may vary.
First, I learn the harmony of the tune inside and out, meaning that I can play it on the bass in all 3 octaves as I hear it. NO SMALL FEAT.
Then, I learn to SING the melody while I play a functional, but maybe 2-4 bass line, and then, the HARD part, I SING the bass line while I play the melody - on bass or piano.
Ah, another step, I play the chords and bass on the piano while I sing the melody. And, I really explore the changes on the piano, trying different voicings until I identify what I "Like" to hear.
Next, I play the chords and bass on the piano while I sing a solo, actually several solos over multiple choruses.
And, I should add, the very first thing I do is locate as many as possible "definitive" recordings of the tune and listen to them non-stop, and then transcribe them enough to determine how they're different.
I play along with the recordings and note how they're different. I think this is important because different players have different recordings in their mind and it's valuable for me to identify which they're hearing. Oh, the pianist went "there", cool!
These steps have gotten me pretty far, IME, but, nearly everyone more advanced than me, encourages me to transcribe and play, verbatim, famous solos, and that's my project for the year, along with improving my ability to transcribe what I hear into notation.
In a nutshell: sing, and transcribe. Singing helps you find your voice; transcribing helps you learn the language.
My two cents.