When the quark model wasn' t there yet, was there evidence that hadrons are composite structures? In hindsight of course yes. But at the time back then, were there signs? Maybe not noticed, but which could have made people thinking this could be the case, without having compositeness in mind a priori?
As @Jon Custer points out, WP covers the history of such models pretty well. Once the classification of hadrons was completed by Gell-Mann and Ne'eman, in 1961, the patterns involved, underlain by flavor SU(3), begged the question of compositeness. A black box rattles and squeals (spectra, couplings) in orderly, non-random ways: this is catnip for theorists. So, discussions by Sakata, and Peterman, succeeded in the end by Gell-Mann and, independently, Zweig's successful introduction of quarks in 1964.
The point is systematic patterns of atomic properties in the periodic table classification of Mendeleev (1870) were ultimately understood by Moseley in 1913 in terms of nuclear charges, building on Rutherford's van den Broeck, and Bohr's work, in a virtually constituent model of the nucleus, let us say. Since then, nuclear physicists cleaned the structure up in detail.
The leap from hadron classification to constituent structure took 3, instead of 45 years, and involved partially the same investigator, MGM, who landed on "mathematical quarks"$^\natural$ through SU(3) group theory (!). George Zweig, a continent apart from MGM, and conceptually so as well, was unique in that he was not driven by expediting SU(3) representation theory, but, instead, dynamics (long life of the φ meson), so he introduced "nuclear physicists' quarks" (constituent quarks; very harshly received at the time: an astounding saga).
For a long time, both types of quarks were highly speculative entities, to be considered or not by the mainstream of the field, certainly not by experimentalists; until, finally, Feynman's parton model (1969) explained scaling through them, and appreciated them as DIS essentially phenomenological objects.
The book by A Pais, Inward Bound, (1986; ISBN13:9780198519713) explains the saga quite wonderfully.
$\natural$ "A search for stable quarks of charge -1/3 or +2/3 and/or stable di-quarks of charge -2/3 or +1/3 or +4/3 at the highest energy accelerators would help to reassure us of the non-existence of real quarks." MGM, 1964