>>1923321It just sounds like you need more cable tension so you should turn the barrell adjuster all the way in, unclamp the cable, pull it tight with pliers, re-clamp, then increase the tension with the adjuster.
If that doesn't work then it's likely that you can adjust it to work correctly if you start from scratch.
>unclamp cable, clean & lightly grease all the threaded pieces, (clamp screw and barrell adjuster). >adjust the derailer position (maybe it is too high or angled) >adjust the limits using your hand to move the derailer>check the shifter is working well with no cable installed if not flush it out and re-lube it >re-cable with a new cable >adjust the tensionIf that doesn't work and setting these up can be tricky, you can install a friction shifter for it. The best ones are the vintage thumbies (deore and xt) with return springs. Then is nice vintage stem shifters (which can be mounted on handlebars, and one can be removed), then is chinky new thumb shifters. Then there's no question of setting the indexing, or cable tension, it just pulls as much cable as you need. And if your current shifter is attached to the brake lever, you can cut it off with a hacksaw and file it smooth.