I don't know the answer to your question, but I asked Fred Cohen. He had this to say:
Most of the computations are in Mahowald's work with the EHP sequence. This gives infinite families at p = 2 with Rob Thompson's extensions to p > 2.
Specific extensions of 2-primary components in fixed stems are in
N. Oda, On the 2-components of the unstable homotopy groups of spheres. II, Proc. Japan Acad. 53, Ser. A(1977), 215-218.
N. Oda, Unstable homotopy groups of spheres, Bull. of the Inst. for Advanced Re- search of Fukuoka Univ. 44 (1979), 49-152.
N. Oda, Some relations in the 18-stem of the homotopy groups of spheres, Bull. Central Res. Inst. Fukuoka Univ., 104, (1988), 75–83.
Brayton Gray gave families of elements of order $p^r$ in $\pi_*(S^{2r+1})$ for $p$ an odd prime.
Example: It is not known whether there are elements of order 64 in $\pi_*(S^{11})$.
Similarly, it is not known whether there are elements of order 64 in the stable homotopy of $\mathbb RP^{10}$, thus a potential counterexample to the Freyd conjecture.
Other than that, not much more in terms of specific computations are in the literature as far as I know.