The limit screws should be easily accessible. Usually they just take a small screw driver or Allen key to adjust and they typically have a small H and L next to it. You should adjust the H screw.

So I linked below what is most likely not the best tires out there but they are a really good deal. A friend of mine recently was fixing up a bike of a similar age and used those them.

https://www.thebikesmiths.com/products/kenda-k35kit?currency=USD&variant=34651753480352&utm_medium=cpc&utm_source=google&utm_campaign=Google Shopping&gclid=Cj0KCQjw7aqkBhDPARIsAKGa0oLxOWiaQZHnAC_o9cxtRGtOv_FgFEkl7c-F2Nwb4co_V4MjKwiLQFwaAlwNEALw_wcB

If this is on the front chain ring I would check your limit screws on your front derailleur to make sure it can actually swing all the way to the position for the third chainring.

Other possible causes, the pawls in the shifter getting gunked up. You could try cleaning it out with WD40 or alcohol. The put back in some oil like tri flow.

Final possibility the cable/housing could be shot and needs to be replaced.