The first published proof that the index of a subgroup of PSL$(2,p)$ is at least $p+1$ for primes $p \ge 13$ is in:

C. Jordan, "Note sur les equations modulaires", C.R. Acad. Sci. Paris 66 (1868), 308-312,

a long time before the classification!

The minimal indexes of subgroups of classical simple groups are determined (again pre-classification) in

B.N. Cooperstein, "Minimal degree for a permutation representation of a classical group", Israel J. Math. 30 (1978), 213-225.

There are apparently a couple of mistakes in Cooperstein's paper, but they concern $U_n(2)$ and orthogonal groups over $F_3$.

In particular, the minimal index of PSL$(n,q)$ is $(q^n-1)/(q-1)$ except for $(n,q)$ = (2,5), (2,7), (2,9), (2,11), or (4,2).