  We consider the family of transcendental entire maps given by
$f_a(z)=a(z-(1-a))\exp(z+a)$ where $a$ is a complex parameter. Every map has a
superattracting fixed point at $z=-a$ and an asymptotic value at $z=0$. For
$a>1$ the Julia set of $f_a$ is known to be homeomorphic to the Sierpi\'nski
universal curve, thus containing embedded copies of any one-dimensional plane
continuum. In this paper we study subcontinua of the Julia set that can be
defined in a combinatorial manner. In particular, we show the existence of
non-landing hairs with prescribed combinatorics embedded in the Julia set for
all parameters $a\geq 3$. We also study the relation between non-landing hairs
and the immediate basin of attraction of $z=-a$. Even as each non-landing hair
accumulates onto the boundary of the immediate basin at a single point, its
closure, nonetheless, becomes an indecomposable subcontinuum of the Julia set.
