There is a proper fix and it costs very little, so do not resort to glue.
A thumbstick cap grips the stick by means of small internal ribs that bite onto a splined or ridged post. The cap is rubber over a plastic core, and over time two things happen. The ribs compress and wear, so the grip loosens, and the rubber itself relaxes slightly with heat and use, which widens the bore. Once the grip falls below a certain point, a quick backward pull is enough to lift it. The left stick goes first because that is the one most people pull back on hardest.
Replacement caps for this controller are a very cheap part, widely available, and they simply push on. A fresh cap grips properly and the problem is gone. Buy a set so you have spares, and replace both sides at the same time, since the right one is the same age and will follow.
What matters more is what not to do. Do not glue it. Adhesive on the splined post fills the grooves that any future cap needs to grip, and removing cured glue from the post without damaging it is genuinely difficult. People do this and then find that no replacement cap will hold either, which turns a two-pound fix into a stick module replacement. The same goes for tape wrapped around the post, which works briefly and then leaves residue.
If you want a temporary fix until a cap arrives, a single turn of thin PTFE plumber's tape around the post is the least harmful option, because it leaves no residue and peels off cleanly. Use that rather than anything adhesive.
One thing to check while you are at it: if the post itself is visibly worn smooth or the splines are rounded, a new cap will not hold either, and that does mean the stick module. That is uncommon at three years but worth a look before you order.
