Tate's isogeny theorem certainly suffices, but one can show much more. Consider the (multi)graph whose vertices are supersingular $j$-invariants mod $p$ and whose edges are isogenies of degree $2$. This family of graphs is not only connected, but it is [Ramanujan][1] \[1, Prop. 3\], meaning that its eigenvalue separation is as large as possible (the graph is as well-connected as possible). Therefore $2$-isogenies alone suffice to reach any supersingular curve from any other supersingular curve. The same statement holds for any other (prime) isogeny degree other than $p$. For example, $37$-isogenies would work (assuming $p \neq 37$).

The proof of the Ramanujan property is too long to repeat here, but essentially the largest eigenvalue is the $p$-th coefficient of the Eisenstein series
$$
\sum_{n=1}^\infty \sigma_1(n) q^n = \sum_{n=1}^\infty \left(\sum_{d \mid n} d^1\right) q^n,
$$
i.e. $p+1$ (which is also the degree of the graph), and the remaining eigenvalues are coefficients of cusp forms of weight $2$ on the modular curve $\Gamma_0(N)$, which are upper bounded by $\sigma_0(p) \sqrt{p} = 2\sqrt{p}$ by the [Ramanujan-Petersson conjecture][2].

\[1\] Pizer, *Ramanujan Graphs and Hecke Operators*, Bulletin of the AMS **23** (1) 1990, pp. 127-137, https://doi.org/10.1090/S0273-0979-1990-15918-X


  [1]: https://en.wikipedia.org/wiki/Ramanujan_graph
  [2]: https://en.wikipedia.org/wiki/Ramanujan%E2%80%93Petersson_conjecture