  Suppose that $\Sigma^n\subset\mathbb{S}^{n+1}$ is a closed embedded minimal
hypersurface. We prove that the first non-zero eigenvalue $\lambda_1$ of the
induced Laplace-Beltrami operator on $\Sigma$ satisfies $\lambda_1 \geq
\frac{n}{2}+ a_n(\Lambda^6 + b_n)^{-1}$, where $a_n$ and $b_n$ are explicit
dimensional constants and $\Lambda$ is an upper bound for the length of the
second fundamental form of $\Sigma$. This provides the first explicitly
computable improvement on Choi & Wang's lower bound $\lambda_1 \geq
\frac{n}{2}$ without any further assumptions on $\Sigma$.
