We investigate the phenomenological aspects of a feebly interacting sterile neutrino dark matter candidate within a low-scale seesaw framework. The Type-I seesaw model is augmented by a second complex scalar doublet ($\Phi_{\nu}$), which couples exclusively to the heavy right-handed neutrinos and the lepton doublet, thereby generating the neutrino Dirac mass term while the first scalar doublet is responsible for giving mass to the remaining Standard Model particles.
The lightest sterile neutrino ($N_1$) acts as a feebly interacting massive particle (FIMP), produced via decays of $W^\pm$, $Z$ and extra scalars present in the setup. We point out that $W^\pm$ and $Z$ contributions were overlooked in the previous studies, which actually dominate the $N_1$ production by a factor of $\sim 10^{13}$ and solely determines the relic abundance. Incorporating them leads to several novel consequences for the DM phenomenology like a new non-thermal condition which leads to smaller Yukawa couplings. We thoroughly discuss the enhancement possibilities of $N_1$'s mass, which is controlled by the small vacuum expectation value ($v_{\nu}$) of the second Higgs doublet. After incorporating the latest Lyman-$\alpha$ forest observations, this setup can accommodate both warm and cold dark matter scenarios. We also discussed the dominant role of SM gauge bosons in dark matter production through heavy-light mixing ($V_{ij} = \frac{M_{D_{ij}}}{M_{N_{j}}}$), which leads to interactions between the heavy right-handed neutrinos with the $W$ and $Z$ bosons. In the context of the FIMP scenario, however, the dark matter couplings are inherently required to be extremely small to remain out-of-equilibrium. So, the out-of-equilibrium condition is the holy grail of the freeze-in mechanism.