  We use Katsura's topological graphs to define Toeplitz extensions of
Latr\'emoli\`ere and Packer's noncommutative-solenoid C*-algebras. We identify
a natural dynamics on each Toeplitz noncommutative solenoid and study the
associated KMS states. Our main result shows that the space of extreme points
of the KMS simplex of the Toeplitz noncommutative torus at a strictly positive
inverse temperature is homeomorphic to a solenoid; indeed, there is an action
of the solenoid group on the Toeplitz noncommutative solenoid that induces a
free and transitive action on the extreme boundary of the KMS simplex. With the
exception of the degenerate case of trivial rotations, at inverse temperature
zero there is a unique KMS state, and only this one factors through
Latr\'emoli\`ere and Packer's noncommutative solenoid.
