  We present a class of cosmological solutions for a generalized teleparallel
gravity with, $f(T)=T+\tilde{\alpha}(-T)^n$, where $\tilde{\alpha}$ is some
parameter and $n$ is an integer or half-integer. Choosing $\tilde{\alpha} \sim
G^{n-1}$, where $G$ is the gravitational constant, and working with an equation
of state $p=w\, \rho$, one obtains a cosmological solution with multiple
branches. The dynamics of the solution describes standard cosmology at late
times, but the higher-torsion correction changes the nature of the initial
singularity from big bang to a sudden singularity. The milder behavior of the
sudden singularity enables us to extend timelike or lightlike curves, through
joining two disconnected branches of solution at the singularity, leaving the
singularity traversable. We show that this extension is consistent with the
field equations through checking the known junction conditions for generalized
teleparallel gravity. This suggests that these solutions describe a contracting
phase a prior to the expanding phase of the universe.
