We show that the presence of a non-contractible one-periodic orbit of a
Hamiltonian diffeomorphism of a connected closed symplectic manifold
$(M,\omega)$ implies the existence of infinitely many non-contractible simple
periodic orbits, provided that the symplectic form $\omega$ is aspherical and
the fundamental group $\pi_1(M)$ is either a virtually abelian group or an
$\mathrm{R}$-group. We also show that a similar statement holds for Hamiltonian
diffeomorphisms of closed monotone or negative monotone symplectic manifolds
under the same conditions on their fundamental groups. These results generalize
some works by Ginzburg and Gürel. The proof uses the filtered Floer--Novikov
homology for non-contractible periodic orbits.