This phase I/IIa trial studies the side effects and best dose of gene-modified T cells when given with or without decitabine, and to see how well they work in treating patients with malignancies expressing cancer-testis antigens 1 (NY-ESO-1) gene that have spread to other places in the body (advanced). A T cell is a type of immune cell that can recognize and kill abnormal cells of the body. Placing a modified gene for NY-ESO-1 into the patients' T cells in the laboratory and then giving them back to the patient may help the body build an immune response to kill tumor cells that express NY-ESO-1. Drugs used in chemotherapy, such as decitabine, work in different ways to stop the growth of tumor cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. It is not yet known whether giving gene-modified T cells with or without decitabine works better in treating patients with malignancies expressing NY-ESO-1.