  In the present paper, we construct two classes of non-weight modules
$\Omega(\lambda,\alpha,\beta)\otimes\mathrm{Ind}(M)$ and
$\mathcal{M}\big(V,\Omega(\lambda,\alpha,\beta)\big)$ over the twisted
Heisenberg-Virasoro algebra, which are both associated with the modules
$\Omega(\lambda,\alpha,\beta)$.
  We present the necessary and sufficient conditions under which modules in
these two classes are irreducible and isomorphic, and also show that the
irreducible modules in these two classes are new. Finally, we construct
non-weight modules $\mathrm{Ind}_{\underline y,\lambda}(\C_{RS})$ and
$\mathrm{Ind}_{\underline z,\lambda}(\C_{PQ})$ over the twisted
Heisenberg-Virasoro algebra and then apply the established results to give
irreducible conditions for $\mathrm{Ind}_{\underline y,\lambda}(\C_{RS})$ and
$\mathrm{Ind}_{\underline z,\lambda}(\C_{PQ})$.
