I've seen a couple of famous bassists over the years advising newer players to focus their learning the fretboard from the nut to the fifth fret. The advice is usually along the lines of "the lowest frets are your 'money makers' - where most producers and bands will want you to play".
Back when I was in an original band, I do recall most of my bass parts used (relied on?) open strings, and I played most of the time around the nut to 5th fret.
Many years later, I'm playing in an accoustic duo now, and I've noticed that most of my playing - of other people's songs, in keys that my guitarist partner is able to sing in - is around the 4th to 8th frets.
I do of course embellish with higher frets from time to time, and also go to lower frets as needed. However, most of my playing seems to be in the zone of the 4th to 8th fret, and I even tend to fret notes on the 5th fret rather than playing open strings.
In which zone do you tend to spend most of your time on the fretboard?
I work in horn bands so there are very few songs where I’m using open strings. Playing by ear, I typically play by intervals of a scale when I’m up the fretboard. Call it the Do-Ray-Me style.
Once I locate the lowest possible root note of a song, positionally everthing is relative fret wise. For instance, going up the scale the 5th note of a major scale is always one string closer the floor, and two frets up the fingerboard, regardless of where the root is located on the fingerboard.
The trick to play like this you need to hear and remember what those intervals sound like. Having relative pitch is useful.
I will also say this works reading standard notation as well. Hope this helps.