  A novel stochastic technique combining a dilute source grid of $\mathbb{Z}_3$
noise with iterative momentum-smearing is used to study the proton correlation
function at rest and in boosted frames on two lattice volumes. The technique
makes use of the baryonic version of the so-called one-end trick, and the
decomposition into signal and noise terms of the resulting stochastic proton
correlation function is made explicit. The number and location of the source
points in the dilute grid should be chosen so that the benefits of averaging
over many locations overcomes the additional statistical error introduced by
the noise terms in the desired fitting region. At all nontrivial momentum
values considered we find that the choice of $N=4$--$8$ maximally separated
source locations is shown to be optimal, providing a reduced statistical error
when compared with a single point source. This enables us to successfully fit
the proton energy at momentum values as high as $|\vec{p}| \simeq 3.75$ GeV and
$|\vec{p}| \simeq 2.82$ GeV on the small and large volume respectively.
