At large momentum transfer, it becomes increasingly difficult to access the form factor of the pion $F_\pi(Q^2)$ using lattice QCD simulations. Two of the limiting factors include the increased computational cost of adding more statistics to overcome gauge noise, as well as suppressed overlap with the ground state of the boosted pion. Here we apply two noise reduction techniques, all-mode averaging (AMA) and momentum smearing, to the computation of $F_\pi(Q^2)$ at high momentum transfers using the Feynman-Hellmann (FH) theorem. First, we show that all-mode averaging by itself produces good improvement compared to previous results, at an equal computational cost. We also implement a momentum smearing technique to further reduce statistical uncertainties. In contrast to conventional smearing approaches, our Feynman-Hellmann method requires combining back-to-back momentum states, and hence we adapt a version of smearing involving a superposition of back-to-back smearing operations. This method is then implemented to compute $F_\pi(Q^2)$ at $Q^2 = 6.6 \;\mathrm{GeV^2}$, demonstrating good improvement over the regular smeared counterpart. Finally both all-mode averaging and momentum smearing are combined to determine $F_\pi(Q^2)$ at $Q^2 = 6.6 \;\mathrm{GeV^2}$ showing an excellent preliminary improvement over previous calculations.