Some families of nearly circular symmetric periodic orbits of comet- and Hill-type in the elliptic restricted three-body problem (ERTBP) are numerically explored by the Broyden method with a linear search. These orbits have inclinations near 90°, where the reference plane is the orbital plane of the primaries. Set the ratio of the mean motions between the orbit of the infinitesimal body and the relative orbit of the primaries as m/k. For the masses of the primaries equal, comet-type doubly-periodic orbitsare calculated when 1/98 ≤ m/k ≤ 1/102. These orbits are not stable for all cases of the initial conditions. Set one mass to be small, some symmetric periodic orbits are also found. For the Hill-type orbits, some symmetric periodic orbits are also calculated withone mass big or small.