Light field (LF) imaging has gained significant attention due to its recent success in 3-dimensional (3D) displaying and rendering as well as augmented and virtual reality usage. Nonetheless, because of the two extra dimensions, LFs are much larger than conventional images. We develop a JPEG-assisted learning-based technique to reconstruct an LF from a JPEG bitstream with a bit per pixel ratio of 0.0047 on average. For compression, we keep the LF's center view and use JPEG compression with 50\% quality. Our reconstruction pipeline consists of a small JPEG enhancement network (JPEG-Hance), a depth estimation network (Depth-Net), followed by view synthesizing by warping the enhanced center view. Our pipeline is significantly faster than using video compression on pseudo-sequences extracted from an LF, both in compression and decompression, while maintaining effective performance. We show that with a 1\% compression time cost and 18x speedup for decompression, our methods reconstructed LFs have better structural similarity index metric (SSIM) and comparable peak signal-to-noise ratio (PSNR) compared to the state-of-the-art video compression techniques used to compress LFs.