The origin of the spiral spin-order in perovskite multiferroic manganites $R$MnO$_{3}$ ($RE=$ Tb or Dy) is here investigated using a two $e_{rm g}$-orbitals double-exchange model. Our main result is that the experimentally observed spiral phase can be stabilized by introducing a relatively weak next-nearest-neighbor superexchange coupling ($sim10%$ of the nearest-neighbor superexchange). Moreover, the Jahn-Teller lattice distortion is also shown to be essential to obtain a realistic spiral period. Supporting our conclusions, the generic phase diagram of undoped perovskite manganites is obtained using Monte Carlo simulations, showing phase transitions from the A-type antiferromagnet, to the spiral phase, and finally to the E-type antiferromagnet, with decreasing size of the $R$ ions. These results are qualitatively explained by the enhanced relative intensity of the superexchanges.
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