  The Large High Altitude Air Shower Observatory (LHAASO) has detected multiple
ultra-high energy (UHE; E$_\gamma \ge$ 100 TeV) gamma-ray sources in the Milky
Way Galaxy, which are associated with Galactic ``PeVatrons'' that accelerate
particles up to PeV (= 10$^{15}$ eV) energies. Although supernova remnants
(SNRs) and pulsar wind nebulae (PWNe), as source classes, are considered the
leading candidates, further theoretical and observational efforts are needed to
find conclusive proof to confirm the nature of these PeVatrons. This work aims
to provide a phenomenological model to account for the emission observed from
the direction of LHAASO J0341+5258, an unidentified UHE gamma-ray source
observed by LHAASO. 15 years of Fermi-LAT data was analyzed to find the high
energy (HE; 100 MeV $\le$ E$_\gamma$ $\le$ 100 GeV) GeV gamma-ray counterpart
of LHAASO J0341+5258, in the 4FGL-DR3 catalog. We have explained the spectrum
of the closest 4FGL source, 4FGL J0340.4+5302, by a synchro-curvature emission
formalism typically used in the case of GeV pulsars. Escape-limited hadronic
interaction between protons accelerated in an old, now invisible SNR and cold
protons inside associated molecular clouds (MCs) and leptonic emission from a
putative TeV halo was explored to explain the multi-wavelength (MWL) spectral
energy distribution (SED) observed from the LHAASO source region. We have
further discussed possible observational avenues that can be explored in the
near future and predicted the outcome of those observational efforts from the
model explored in this paper.
