I think it depends on how flat the quilt is now. I've washed quilts and found that sometimes the unquilted parts seem to have "puffy" (I can't think of another word for the loose fabric) areas that would cause the quilt to have lumpy areas when quilted more densely. I've found this difficult to deal with and had decided not to quilt in those areas. Since I now glue the fabric to the batting, I don't have that problem anymore. However, I do a lot more quilting on my quilts as my skills have improved over the years.
I'd suggest that you try quilting in a small area and see how you like the results. Sometimes I'm must more picky about what "will do" versus must be "good enough" to give away.