Here's a great exercise for building up fretting hand strength and stretches on a 5 string bass guitar. If you play a 4 string bass guitar that's no dramas, you can still do this exercise. Just ignore the bottom string, the B-string.
This exercise is an e minor pentatonic scale played in triplets. The last note of each triplet is also the first note of the next triplet. Using note numbers that would be 1-2-3, 3-4-5, 5-6-7, etc. or in note names it would be B-D-E, E-G-A, A-B-D, etc. The point of this lesson isn't about learning the notes but that will certainly help in applying this exercise to your bass playing.
You will also notice that the scale is played three notes per string. Melodically, it sounds great, but doesn't exactly emphasize each note equally. Physically, it's a great way to build up hand strength and accuracy. The fretting hand fingering given in blue is probably the most logical way to play this exercise. Try substituting the red notes (second and pinky fingers) on the strings shown for a challenge. That makes the playing approach strictly one-finger-per-fret and makes the exercise a bit more demanding.
Both ways of playing are equally valid and your choice may come down to personal preference. If you implement something like this into your playing style genre and song style may influence your fretting hand approach. In your practice sessions use both versions to give your whole hand a workout.