  We present a new family of exact four-dimensional Taub-NUT spacetimes in
Einstein-$\Lambda$ theory supplemented with a conformally coupled scalar field
exhibiting a power-counting super-renormalizable potential. The construction
proceeds as follows: A solution of a conformally coupled theory with a
conformal potential, henceforth the seed $(g_{\mu\nu},\phi)$, is transformed by
the action of a specific change of frame in addition with a simultaneous shift
of the seed scalar. The new configuration, $(\bar{g}_{\mu\nu},\bar{\phi})$,
solves the field equations of a conformally coupled theory with the
aforementioned super-renormalizable potential. The solution spectrum of the
seed is notoriously enhanced. We highlight the existence of two types of exact
black bounces given by de Sitter and anti-de Sitter geometries that transit
across three different configurations each. The de Sitter geometries transit
from a regular black hole with event and cosmological horizons to a bouncing
cosmology connecting two de Sitter Universes with different values of the
asymptotic cosmological constant. An intermediate phase represented by a de
Sitter wormhole or by a bouncing cosmology that connects two de Sitter
Universes is shown, this under the presence of a cosmological horizon. On the
other hand, the anti-de Sitter geometries transit from a regular black hole
with inner and event horizons to a wormhole that connects two asymptotic
boundaries with different constant curvatures. The intermediate phase is given
by an anti-de Sitter regular black hole with a single event horizon that
appears in two different settings. As a regular anti-de Sitter black hole
inside of an anti-de Sitter wormhole or as an anti-de Sitter regular black hole
with an internal cosmological bounce. These geometries are smoothly connected
by the mass parameter only. Other black holes, bouncing cosmologies and
wormholes are also found.
