In any radioactive series we can observe that the half-lives of a daughter nucleus might be less than that of the parent nucleus. If the reason for radioactivity is gaining stability, why does a more stable nucleus (the one with higher half-life) turn into a less stable (the one with smaller half-life value) one?
As you say, the daughter nucleus may be less stable, therefore the reason for radioactivity cannot be to gain stability.
Instability simply means that there exists an allowed transition to a lower energy state. For a very unstable state that transition typically takes a short amount of time. For a completely stable state that transition takes an infinite amount of time. Therefore, once it gets to a stable state then it will not decay further, but the “reason” for the decays is the loss of energy rather than the gain in stability.
Stability is more of a stopping criterion than a driving force.
There are several reasons why that might be true. Here is one, and I invite the experts here to weigh in with more.
We know from experiment that there are several different fundamental physical processes that can take place during radioactive decay. They have different characteristic time scales that in turn depend on things like conservation laws, which allow some decay mechanisms while prohibiting others at each step in the overall decay process.
So an unstable nucleus may begin its path towards greater stability with one particular decay mechanism that is relatively slow, and then follow it with a second step that is faster, and a third which might be slower, and so on.