If future neutrino oscillation experiments show that the neutrino mass spectrum is with normal ordering, m1 < m2 < m3, and the searches for neutrinoless double beta-decay with sensitivity to values of the effective Majorana mass |m| > 0.01 eV give negative results, the next frontier in the quest for neutrinoless double beta-decay will correspond to |m| ~ 0.001 eV. Assuming that massive neutrinos are Majorana particles and their exchange is the dominant mechanism generating neutrinoless double beta-decay, we analise the conditions under which |m|, in the case of three neutrino mixing and neutrino mass spectrum with normal ordering, would satisfy |m| > 0.001 eV. We consider the specific cases of i) normal hierarchical neutrino mass spectrum, ii) of relatively small value of the CHOOZ angle theta13 as well as iii) the general case of spectrum with normal ordering, partial hierarchy and a value of theta13 close to the existing upper limit. We study the ranges of the lightest neutrino mass m1 and/or of sin^2 theta13, for which |m|> 0.001 eV and discuss the phenomenological implications of such scenarios. We provide also an estimate of |m| when the three neutrino masses and the neutrino mixing originate from neutrino mass term of Majorana type for the (left-handed) flavour neutrinos and m1 Ue1^2 + m2 U_e2^2 + m3 Ue3^2 =0, but there does not exist a symmetry which forbids the neutrinoless double beta-decay.