<p>Holographic thermal two-point functions can be analyzed using the operator product expansion which contains contributions from both multi-stress-tensor and double-trace operators. The former can be computed by analyzing the bulk equation of motion in a near-boundary expansion, but the latter has remained elusive — in practice, one resorts to solving a partial differential equation with limited accuracy. We show that imposing the Euclidean periodicity condition on the holographic correlator (also known as the KMS condition or thermal bootstrap), followed by Padé-Borel resummation, provides an efficient method for computing double-trace thermal coefficients. The resulting series converges rapidly and yields numerical values in excellent agreement with those obtained from solving the partial differential equation.</p>