  Completing a previous analysis started in [1], we study flat
Friedmann--Lema\^{\i}tre--Robertson--Walker (FLRW) models with a perfect fluid
matter source and a scalar field nonminimally coupled to matter,
self--interacting with a potential that may attain negative values. We prove
that the evolution generically forces the Hubble function to diverge to
$-\infty$ in a finite time, except in case the potential exhibits a flat
plateau at infinity (tending to zero from below); in that case we find
conditions which may give rise to ever expanding or recollapsing cosmologies.
