We consider a generalized model of random walk in dynamical random environment, and we show that the multiplicative-noise stochastic heat equation (SHE) describes the fluctuations of the quenched density at a certain precise location in the tail. The distribution of transition kernels is fixed rather than changing under the diffusive rescaling of space-time, i.e., there is no critical tuning of the model parameters when scaling to the stochastic PDE limit. The proof is done by pushing the methods developed in [arxiv 2304.14279, arXiv 2311.09151] to their maximum, substantially weakening the assumptions and obtaining fairly sharp conditions under which one expects to see the SHE arise in a wide variety of random walk models in random media. In particular we are able to get rid of conditions such as nearest-neighbor interaction as well as spatial independence of quenched transition kernels. Moreover, we observe an entire hierarchy of moderate deviation exponents at which the SHE can be found, confirming a physics prediction of J. Hass.