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, 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.

[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

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, 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 $\ell$ 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 of the $\ell$-isogeny graph is the $\ell$-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. $\ell+1$ (which is also the degree of the graph), and the remaining eigenvalues are coefficients of cusp forms of weight $2$ on the modular curve $\Gamma_0(N)$ for some suitably chosen $N$, which are upper bounded by $\sigma_0(\ell) \sqrt{\ell} = 2\sqrt{\ell}$ by the Ramanujan-Petersson conjecture.

[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