Getting into 3rd or not can be extremely sensitive on a too high cable tension
Others mentioned cable tension, and you'd think that is obvious, but I just wanted to highlight that quite small changes in cable tension can make a huge difference.
For example, here's a picture of my Shimano ST-EF65 opened up:
By pushing the thumb lever, I could make the chain go into 3rd. But I could not for the life in me get it to click into 3rd inside the shifter and stay there, and so the shifter would fall back to 2nd.
It had always been a bit hard, but something happened and it became impossible! Here's me trying to do it with my entire hand instead of just the thumb as you would in riding position:
After trying a bunch of things, including spraying WD-40 into the shifter to try and remove old lubricant as suggested in many videos, I finally played with the tension a bit further, and only then got it to work.
Once I was in a working state, I noticed that simply turning the barrel adjuster 5 quarter turns already made the difference between being able to shift or not!
The lesson is: the mechanical strengths and precisions of a cable can be really surprising. You'd think that your thumb is really strong and would be able to overcome them with the brute strength of or hands, but that may not be true.
Lube the front derailleur
Soon after getting it to work silky smooth as mentioned previously, and including cable changes, I went for a 4 hour ride on wet conditions, and I could clearly notice how progressively the 3rd was again getting harder and harder to get into, until it got impossible again.
Since the cable tension had been unchanged during the ride, I concluded that the main factor now was actually the front derailleur, which is close to the bottom bracket, and so that is the most likely part where water would have gotten into, as opposed to the shifter.
What must have happened is that the WD-40 I had previously put on just washed off during the wet ride, as it is not a proper lube, mostly a water displacer.
So what I did was:
- put WD-40 again on the hinges. I then managed once again to go into 3rd.
- let it dry off overnight
- put on some actual spray lube, "WD-40 Bike, All Conditions Lube Bicycle Lubricant, 250ml"
After doing this, it has been holding for several weeks, including after a few wet rides, so after all, that seemed to be the key issue.
Part of the difficulty of this diagnosis is that the WD-40 doesn't take effect immediately it seems, you have to move the hinges around several times to let it penetrate and remove what as creating the friction.
Related question: Very stiff front derailleur
These are the three hinges I focused on lubing on both sides: