  We propose a new methodology aimed at finding star-forming galaxies in the
phase which immediately follows the star-formation (SF) quenching, based on the
use of high- to low-ionization emission line ratios. These ratios rapidly
disappear after the SF halt, due to the softening of the UV ionizing radiation.
We focus on [O III] $\lambda$5007/H$\alpha$ and [Ne III] $\lambda$3869/[O II]
$\lambda$3727, studying them with simulations obtained with the CLOUDY
photoionization code. If a sharp quenching is assumed, we find that the two
ratios are very sensitive tracers as they drop by a factor $\sim$ 10 within
$\sim$ 10 Myr from the interruption of the SF; instead, if a smoother and
slower SF decline is assumed (i.e. an exponentially declining star-formation
history with $e$-folding time $\tau=$ 200 Myr), they decrease by a factor
$\sim$ 2 within $\sim$ 80 Myr. We mitigate the ionization -- metallicity
degeneracy affecting our methodology using pairs of emission line ratios
separately related to metallicity and ionization, adopting the [N II]
$\lambda$6584/[O II] $\lambda$3727 ratio as metallicity diagnostic. Using a
Sloan Digital Sky Survey galaxy sample, we identify 10 examples among the most
extreme quenching candidates within the [O III] $\lambda$5007/H$\alpha$ vs. [N
II] $\lambda$6584/[O II] $\lambda$3727 plane, characterized by low [O III]
$\lambda$5007/H$\alpha$, faint [Ne III] $\lambda$3869, and by blue
dust-corrected spectra and $(u-r)$ colours, as expected if the SF quenching has
occurred in the very recent past. Our results also suggest that the observed
fractions of quenching candidates can be used to constrain the quenching
mechanism at work and its time-scales.
