  A simple elliptic singularity of type $E_N^{(1,1)}$ ($N=6,7,8$) can be
described in terms of a marginal deformation of an invertible polynomial $W$.
In the papers \cite{KS} and \cite{MR} the authors proved a mirror symmetry
statement for some particular choices of $W$ and used it to prove
quasi-modularity of Gromov-Witten invariants for certain elliptic orbifold
$\mathbb{P}^1$s. However, the choice of the polynomial $W$ and its marginal
deformation $\phi_{\mu}$ are not unique. In this paper, we investigate the
global mirror symmetry phenomenon for the one-parameter family
$W+\sigma\phi_{\mu}$. In each case the mirror symmetry is governed by a certain
system of hypergeometric equations. We conjecture that the Saito-Givental
theory of $W+\sigma\phi_{\mu}$ at any special limit $\sigma$ is mirror to
either the Gromov-Witten theory of an elliptic orbifold $\mathbb{P}^1$ or the
Fan-Jarvis-Ruan-Witten theory of an invertible simple elliptic singularity with
diagonal symmetries, and the limits are classified by the Milnor number of the
singularity and the $j$-invariant at the special limit. We prove the conjecture
when $W$ is a Fermat polynomial. We also prove that the conjecture is true at
the Gepner point $\sigma=0$ in all other cases.
