Hi Laura,
Can't help you with a FREE doc to epub program, but I believe the Atlantis Word Processor will read DOC and write EPUB for a reasonable price ($35 I think). I haven't used it myself but I believe you can try it for free.
https://www.mobileread.com/forums/showthread.php?t=48775
(Sorry, cross-post ...)
If you decide to stick with the long-winded-but-free option of MSWord I would recommend you convert your DOCs to HTML(Webpage-Filtered) rather than RTF before importing to Calibre. I think you will find more of the formatting in your source DOC will be correctly converted to the EPUB e.g. centre, right-align, hard-line-breaks to name but a few.