We study a simple extension of the Standard Model supplemented by an electroweak triplet scalar field to accommodate small neutrino masses by the type-II seesaw mechanism, while an additional singlet scalar field can play the role of cold dark matter (DM) in our Universe. This DM candidate is leptophilic for a wide range of model parameter space, and the lepton flux due to its annihilation carries information about the neutrino mass hierarchy. Using the recently released high-precision data on positron fraction and flux from the AMS-02 experiment, we examine the DM interpretation of the observed positron excess in our model for two kinematically distinct scenarios with the DM and triplet scalar masses (a) nondegenerate (m⌄(DM) ≫ m⌄Δ) and (b) quasidegenerate (m⌄(DM) ≃ m⌄ Δ). We find that a good fit to the AMS-02 data can be obtained in both cases (a) and (b) with a normal hierarchy of neutrino masses, while the inverted hierarchy case is somewhat disfavored. Although we require a larger boost factor for the normal hierarchy case, this is still consistent with the current upper limits derived from Fermi-LAT and IceCube data for case (a). Moreover, the absence of an excess antiproton flux as suggested by PAMELA data sets an indirect upper limit on the DM-nucleon spin-independent elastic scattering cross section, which is stronger than the existing DM direct detection bound from LUX in the AMS-02 preferred DM mass range.