  Detections of extended Ly$\alpha$ halos (LAHs) around Ly$\alpha$ emitters
(LAEs) have lately been reported on a regular basis, but their origin is still
under investigation. Simulation studies predict that the outer regions of the
extended LAHs contain a major contribution from the Ly$\alpha$ emission of
faint, individually undetected LAEs. To address this matter from an
observational angle, we use halo occupation distribution (HOD) modeling to
reproduce the clustering of a spectroscopic sample of 1265 LAEs at $3<z<5$ from
the MUSE-Wide survey. We integrate the Ly$\alpha$ luminosity function (LF) to
estimate the background surface brightness due to discrete faint LAEs. We then
extend the HOD statistics inwards towards small separations and compute the
factor by which the measured Ly$\alpha$ surface brightness (SB) is enhanced by
undetected close physical neighbors. We consider various clustering scenarios
for the undetected sources and compare the corresponding radial profiles. The
resulting inferred Ly$\alpha$ SB of faint LAEs ranges between
$(0.4-2)\times10^{20}\;\rm{erg}\;\rm{s}^{-1}~\rm{cm}^{-2}~\rm{arcsec}^{-2}$,
with a very slow radial decline outwards. Our results suggest that the outer
regions of observed LAHs ($R\gtrsim50$~pkpc) could indeed contain a strong
component from external (but physically associated) LAEs, possibly even be
dominated by them. Only for a relatively shallow faint-end slope of the
Ly$\alpha$ LF would this contribution from clustered LAEs become unimportant.
We also confirm that the observed emission from the inner regions
($R\le20-30$~pkpc) is too bright to be significantly affected by clustering. We
compare our findings with predicted profiles from simulations and find good
overall agreement. We outline possible future measurements to further constrain
the impact of discrete undetected LAEs on observed extended LAHs.
