  We propose a new model of scalarized neutron stars (NSs) realized by a
self-interacting scalar field $\phi$ nonminimally coupled to the Ricci scalar
$R$ of the form $F(\phi)R$. The scalar field has a self-interacting potential
and sits at its vacuum expectation value $\phi_v$ far away from the source.
Inside the NS, the dominance of a positive nonminimal coupling over a negative
mass squared of the potential leads to a symmetry restoration with the central
field value $\phi_c$ close to $0$. This allows the existence of scalarized NS
solutions connecting $\phi_v$ with $\phi_c$ whose difference is significant,
whereas the field is located in the vicinity of $\phi=\phi_v$ for weak
gravitational stars. The Arnowitt-Deser-Misner mass and radius of NSs as well
as the gravitational force around the NS surface can receive sizable
corrections from the scalar hair, while satisfying local gravity constraints in
the Solar system. Unlike the original scenario of spontaneous scalarization
induced by a negative nonminimal coupling, the catastrophic instability of
cosmological solutions can be avoided. We also study the cosmological dynamics
from the inflationary epoch to today and show that the scalar field $\phi$
finally approaches the asymptotic value $\phi_v$ without spoiling a successful
cosmological evolution. After $\phi$ starts to oscillate about the potential
minimum, the same field can also be the source for cold dark matter.
