We consider the family of transcendental entire maps given by $f_a\left(z\right)=a(z-(1-a))exp(z+a)$ 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 Sierpiński 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...