An isoperimetric inequality on the Hamming cube for exponents $β\ge 0.50057$ is proved, achieving equality on any subcube. This was previously known for $β\ge \log_2(3/2)\approx 0.585$. Improved bounds are also obtained at the critical exponent $β=0.5$, including a bound that is asymptotically sharp for small subsets. A key ingredient is a new Bellman-type function involving the Gaussian isoperimetric profile which appears to be a good approximation of the true envelope function. Verification uses computer-assisted proofs and interval arithmetic. Applications include progress towards a conjecture of Kahn and Park as well as sharp Poincaré inequalities for Boolean-valued functions near $L^1$.