  We revisit the abstract framework underlying the fibration method for
producing rational points on the total space of fibrations over the projective
line. By fine-tuning its dependence on external arithmetic conjectures, we
render the method unconditional when the degree of the non-split locus is $\leq
2$, as well as in various instances where it is $3$. We are also able to obtain
improved results in the regime that is conditionally accessible under
Schinzel's hypothesis, by incorporating into it, for the first time, a
technique due to Harari for controlling the Brauer--Manin obstruction in
families.
