  It is known that, for many transcendental entire functions in the
Eremenko-Lyubich class $\mathcal{B}$, every escaping point can eventually be
connected to infinity by a curve of escaping points. When this is the case, we
say that the functions are criniferous. In this paper, we extend this result to
a new class of maps in $\mathcal{B}$. Furthermore, we show that if a map
belongs to this class, then its Julia set contains a Cantor bouquet; in other
words, it is a subset of $\mathbb{C}$ ambiently homeomorphic to a straight
brush.
