In order to unlock the full advantages of massive multiple input multiple output (MIMO) in the downlink, channel state information (CSI) is required at the base station (BS) to optimize the beamforming matrices. In frequency division duplex (FDD) systems, full channel reciprocity does not hold, and CSI acquisition generally requires downlink pilot transmission followed by uplink feedback. Prior work proposed the end-to-end design of pilot transmission, feedback, and CSI estimation via deep learning. In this work, we introduce an enhanced end-to-end design that leverages partial uplink-downlink reciprocity and temporal correlation of the fading processes by utilizing jointly downlink and uplink pilots. The proposed method is based on a novel deep learning architecture - HyperRNN - that combines hypernetworks and recurrent neural networks (RNNs) to optimize the transfer of long-term channel features from uplink to downlink. Simulation results demonstrate that the HyperRNN achieves a lower normalized mean square error (NMSE) performance, and that it reduces requirements in terms of pilot lengths.