Shift-map codes have been studied as joint source-channel codes for continuous sources. These codes are useful in delay-limited scenarios and also provide better tolerance to deviations of the signal-to-noise ratio (SNR) from a target SNR, compared to separate source and channel coding. This paper defines a generalized family of shift-map codes that share a strong connection with redundant residue number systems (RRNS), and are henceforth called RRNS-map codes. In the proposed coding scheme, side information about the source allows the decoder to consider only a fraction of the codebook for decoding, with no change in the encoding process. With an appropriately designed RRNS-map code, in this fraction of the codebook, the codewords are much better separated than the original codebook. As a result, RRNS-map codes achieve the same distortion in the mean square error sense as conventional shift-map codes without side information, but significantly outperform shift-map codes when side information is provided to the decoder. This coding scheme is ideally suited to applications where a simple and fixed encoding scheme is desired at the encoder, while the decoder is given access to side information about the source.