  Identifying the Milky Way's very high energy hadronic cosmic-ray accelerators
-- the PeVatrons -- is a critical problem. While gamma-ray observations reveal
promising candidate sources, neutrino detection is needed for certainty, and
this has not yet been successful. Why not? There are several possibilities, as
we delineated in a recent paper [T. Sudoh and J. F. Beacom, Phys. Rev. D 107,
043002 (2023)]. Here we further explore the possibility that the challenges
arise because PeVatrons have a large angular extent, either due to cosmic-ray
propagation effects or due to clusters of sources. We show that while extended
neutrino sources could be missed in the commonly used muon-track channel, they
could be discovered in the all-flavor shower channel, which has a lower
atmospheric-neutrino background flux per solid angle. Intrinsically, showers
are quite directional and would appear so in water-based detectors like the
future KM3NeT, even though they are presently badly smeared by light scattering
in ice-based detectors like IceCube. Our results motivate new shower-based
searches as part of the comprehensive approach to identifying the Milky Way's
hadronic PeVatrons.
