Unfortunately, cells that need to undergo meiosis have already undergone this 'S' phase duplication of their genes. Remember that the ultimate aim of meiosis is to produce a cell with half the original genetic material - so that it can fuse with a gamete of the opposite sex (e.g. egg cell) and together reconstitute the full amount of genes.
You will see then, that two divisions are needed to accomplish this, since the amount of genetic material in a cell entering meiosis is already double the normal (two sister chromatids for each chromosome). The first division (meiosis I) splits the male- and female-derived chromosomes apart (you have two chromosomes 1, 2, 3, etc., one from each parent). The second division splits the sister chromatids of each chromosome apart, almost like in conventional mitosis.