In this paper we propose the Iterative Amortized Hierarchical Variational Autoencoder (IA-HVAE), which expands on amortized inference with a hybrid scheme containing an initial amortized guess and iterative refinement with decoder gradients. We achieve this by creating a linearly separable decoder in a transform domain (e.g. Fourier space), enabling real-time applications with very high model depths. The architectural change leads to a 35x speed-up for iterative inference with respect to the traditional HVAE. We show that our hybrid approach outperforms fully amortized and fully iterative equivalents in accuracy and speed respectively. Moreover, the IAHVAE shows improved reconstruction quality over a vanilla HVAE in inverse problems such as deblurring and denoising.