Re-ripping is the only option if you want a perfect copy of the original music.
Converting to FLAC from mp3 will not be helpful: the resulting files will be, at best, as good as your source mp3s, and more likely worse. This is because mp3 is a lossy form of encoding, it discards information from the original disc as part of the encoding process. Once that information is gone, it's gone; you can't reconstitute it algorithmically.
IF you rip to FLAC the encoder losslessly encodes everything from the original disc. With a FLAC rip you can later convert to other formats with no loss because the FLAC file has all the information that was on the original disc. So if you were asking this question in reverse (i.e. can I convert FLAC to mp3 or do I need to re-rip) conversion would be perfectly fine. That's the main advantage of FLAC in my opinion (it's an ideal archival format).