  Finding a method to combine the numerical bootstrap with the analytic
lightcone bootstrap is an important goal to advance the conformal bootstrap
program. We propose a hybrid bootstrap method to do just that. The numerical
and analytic bootstrap approaches are sensitive to different regions of the
spectrum and complement each other. When they are effectively combined, the
hybrid bootstrap enjoys the best of both worlds and the prediction for the
actual CFT can be significantly improved. In this work, we discuss the general
strategy to perform such a hybrid bootstrap, and we make a partial
implementation of the strategy for 3D Ising CFT $\{\sigma,\epsilon\}$ system.
Even at relatively low derivative order $\Lambda=19$, the hybrid bootstrap
predicts very precise values for the scaling dimension
$\Delta_\sigma,\Delta_\epsilon$ that are within the previous $\Lambda=43$
rigorous error bars.
