Gene-Modified T Cells With or Without Decitabine in Treating Patients With Advanced Malignancies Expressing NY-ESO-1

Inclusion Criteria: * Patients with solid tumors as described below: * Inoperable or metastatic (advanced) melanoma: * Has received, is intolerant, or refused a CTLA-4 inhibitor (ipilimumab) or a PD-1 inhibitor (nivolumab or pembrolizumab) as monotherapy and/or a combination of ipilimumab and nivolumab * Has received or is intolerant of a BRAF inhibitor or the combination of BRAF and MEK inhibitors for BRAFv600 mutant melanoma and a PD-1 inhibitor as monotherapy or in combination * Inoperable or metastatic (advanced) ovarian, primary peritoneal or fallopian tube carcinoma: * Has received platinum containing chemotherapy and has platinum refractory or resistant disease that has progressed on second line therapy * If platinum sensitive disease, should have received \>= 2 lines of chemotherapy * May have received PARP inhibitors, bevacizumab or other targeted VEGF inhibitor therapy * Inoperable or metastatic (advanced) synovial sarcoma: * Should have received and progressed on \>= two lines of systemic therapy * Subjects with other histologies: * Must have previously received two lines of systemic standard care (or effective salvage chemotherapy regimens) for metastatic disease, if known to be effective for that disease, and have been deemed either non-responders (progressive disease) or have recurred * For cohorts 1, 2 and 3 only: Patient's tumor must be positive by histological or molecular assay for NY-ESO-1, according to the screenin…