  Given a combinatorial design $\mathcal{D}$ with block set $\mathcal{B}$, the
block-intersection graph (BIG) of $\mathcal{D}$ is the graph that has
$\mathcal{B}$ as its vertex set, where two vertices $B_{1} \in \mathcal{B}$ and
$B_{2} \in \mathcal{B} $ are adjacent if and only if $|B_{1} \cap B_{2}| > 0$.
The $i$-block-intersection graph ($i$-BIG) of $\mathcal{D}$ is the graph that
has $\mathcal{B}$ as its vertex set, where two vertices $B_{1} \in \mathcal{B}$
and $B_{2} \in \mathcal{B}$ are adjacent if and only if $|B_{1} \cap B_{2}| =
i$. In this paper several constructions are obtained that start with twofold
triple systems (TTSs) with Hamiltonian $2$-BIGs and result in larger TTSs that
also have Hamiltonian $2$-BIGs. These constructions collectively enable us to
determine the complete spectrum of TTSs with Hamiltonian $2$-BIGs (equivalently
TTSs with cyclic $2$-intersecting Gray codes) as well as the complete spectrum
for TTSs with $2$-BIGs that have Hamilton paths (i.e., for TTSs with
$2$-intersecting Gray codes).
  In order to prove these spectrum results, we sometimes require ingredient
TTSs that have large partial parallel classes; we prove lower bounds on the
sizes of partial parallel clasess in arbitrary TTSs, and then construct larger
TTSs with both cyclic $2$-intersecting Gray codes and parallel classes.
