Also just to make sure we’re still talking the same version of English, your problem is only after you have started and driven the car? (Meaning - when it’s warmed up..)
Or does it also not start after it sat over night also?
And - you mentioned parking on a hill so you could start it via the ol’ pop the clutch trick. Was that in conjunction with this issue or only when the starter was acting up? And if with the present issue, does it start when you do that? (’Cause that would really wander into a strange and mysterious land..)
The first time I start it, it starts like it always had in the past. The second and/or third or more times, it is a real problem child.
I’ve always used the clutch-start method when I haven’t been able to get the engine to turn over, either before the starter was replaced or with battery problems (my old Toyota truck) and though I park on a slight incline, I prefer to start it with the key.
And so far, with this issue, I’ve only had to do it once, but it was like my third or fourth stop during errand-running.