Let $A = [0,1] \times \mathbb{S}^1$ be the standard annulus and let $X$ be the space obtained from $A$ by identifying $\{0\} \times \mathbb{S}^1$ and $\{1\} \times \mathbb{S}^1$ by a map which represents twice the generator of $\pi_1(\mathbb{S}^1)$.
I am looking for a hint on how to find the $\Delta-$complex structure (cf Hatcher) of $X$.
The first hint is semantic: to write of the $\Delta$ complex structure on $X$ is an error. There are many different $\Delta$ complex structures on $X$, and your job is to construct just one of them.
And here's a hint to the construction: first choose vertices on $\partial A = \{0,1\} \times \mathbb S^1$ so that the identification map restricts to a two-to-one surjection from the chosen vertices on $\{0\} \times \mathbb S^1$ to the chosen vertices on $\{1\} \times \mathbb S^1$.