To this I might add a rhetorical question: how would science have gone about discovering the scientific method? Indeed, in which discipline would that question fall even today?
The general problem of whether we can increase the probability that a theory is true by observing favorable evidence is the problem of induction, and there are no clear cut answers. A lot of philosophy focuses on situations like this: where all the proposed answers have flaws, but some have fewer flaws than others.
For example, one approach that's been gaining popularity is Bayesianism. Bayesianism is the idea that we have some prior belief in the probability of a hypothesis, and that we can update our confidence in that hypothesis by observing evidence and comparing the relative probabilities that evidence would hold in worlds where the hypothesis was true vs. worlds where the hypothesis was false. The key weakness of Bayesianism is the question of where priors come from, and how to formulate good priors. This is a glaring weakness, but all proposed solutions have glaring weaknesses–that's what makes Philosophy of Science hard.