By following Ryan's leads I've been able to find references computing $\pi_{n+k}(S^n)$ for $20\leq k \leq 30$ at the prime $2$ (and in some cases at odd primes as well). I thought I'd post these as an answer for ease of reference.
- M. Mimura, H. Toda, The (n+20)-th homotopy groups of n-spheres, J. Math. Kyoto Univ. 3 1963 37–58. [Contains $\pi_{n+20}(S^n)$ for all $n$ and at all primes]
- M. Mimura, On the generalized Hopf homomorphism and the higher composition, Parts I, II. J. Math. Kyoto Univ., 4, 171-190, 301-326 (1964/5). [Contains $\pi_{n+21}(S^n)$ and $\pi_{n+22}(S^n)$ for all $n$ and at all primes. For odd primes the author refers to work of Toda later published as a series of papers On iterated suspensions I--IV, J. Math. Kyoto Univ (?1966/68).]
- M. Mimura, M. Mori, N. Oda, Determination of 2-components of the 23- and 24-stems in homotopy groups of spheres, Mem. Fac. Sci. Kyushu Univ. Ser. A 29 (1975), no. 1, 1–42. [Contains $\pi_{n+23}(S^n)$ and $\pi_{n+24}(S^n)$ for all $n$ at the prime $2$]
- N. Oda, On the 2-components of the unstable homotopy groups of spheres Parts I, II, Proc. Japan Acad. 53, Ser. A(1977), no. 6/7, 202-205/215-218. [Contains $\pi_{n+k}(S^n)$ for $25\leq k\leq 30$ for all $n$ at the prime $2$, and partial results for $31\leq k\leq 33$]
The methods are as in Toda's book. I would be interested to learn of any later references, or errata to the above.