It has been hypothesized that $k$-SAT is hard to solve for randomly chosen instances near the "critical threshold", where the clause-to-variable ratio is $2^k \ln 2-θ(1)$. Feige's hypothesis for $k$-SAT says that for all sufficiently large clause-to-variable ratios, random $k$-SAT cannot be refuted in polynomial time. It has also been hypothesized that the worst-case $k$-SAT problem cannot be solved in $2^{n(1-ω_k(1)/k)}$ time, as multiple known algorithmic paradigms (backtracking, local search and the polynomial method) only yield an $2^{n(1-1/O(k))}$ time algorithm. This hypothesis has been called the "Super-Strong ETH", modeled after the ETH and the Strong ETH. Our main result is a randomized algorithm which refutes the Super-Strong ETH for the case of random $k$-SAT, for any clause-to-variable ratio. Given any random $k$-SAT instance $F$ with $n$ variables and $m$ clauses, our algorithm decides satisfiability for $F$ in $2^{n(1-Ω(\log k)/k)}$ time, with high probability. It turns out that a well-known algorithm from the literature on SAT algorithms does the job: the PPZ algorithm of Paturi, Pudlak, and Zane (1998).