  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. 
AdS-Taub-NUT spacetimes and exact black bounces with scalar hair