  The interaction between the ejecta of supernovae (SNe) of Type IIn and a
dense circumstellar medium (CSM) can efficiently generate thermal UV/optical
radiation and lead to the emission of neutrinos in the $1$-$10^{3}$~TeV range.
We investigate the connection between the neutrino signal detectable at the
IceCube Neutrino Observatory and the electromagnetic signal observable by
optical wide-field, high-cadence surveys to outline the best strategy for
upcoming follow-up searches. We outline a semi-analytical model that connects
the optical lightcurve properties to the SN parameters and find that a large
peak luminosity (${L_{\rm{peak}}\gtrsim 10^{43}-10^{44}\,\mathrm{erg
\,s^{-1}}}$) and an average rise time ($t_{\rm{rise}}\gtrsim 10-40$~days) are
necessary for copious neutrino emission. Nevertheless, the most promising $
L_{\rm{peak}}$ and $t_{\rm{rise}}$ can be obtained for SN configurations that
are not optimal for neutrino emission. Such ambiguous correspondence between
the optical lightcurve properties and the number of IceCube neutrino events
implies that relying on optical observations only, a range of expected neutrino
events should be considered (e.g.~the expected number of neutrino events can
vary up to two orders of magnitude for some among the brightest SNe IIn
observed by the Zwicky Transient Facility up to now, SN 2020usa and SN 2020in).
In addition, the peak in the high-energy neutrino curve should be expected a
few $t_{\rm{rise}}$ after the peak in the optical lightcurve. Our findings
highlight that it is crucial to infer the SN properties from multi-wavelength
observations rather than focusing on the optical band only to enhance upcoming
neutrino searches.
