TCR Genetically Engineered PBMC and PBSC After Melphalan Conditioning Regimen in Treating Participants With Relapsed and Refractory Multiple Myeloma

Inclusion Criteria: * Relapsed, relapsed and refractory or refractory multiple myeloma patients who have received \> 3 prior lines of therapy including a proteasome inhibitor, an immunomodulatory agent, and an anti-CD28 monoclonal antibody * NY-ESO-1 positive by immunohistochemistry (IHC) utilizing commercially available NY-ESO-1 antibodies * HLA-A\*0201 (HLA-A2.1) positivity by molecular subtyping * Measurable disease defined by at least one of the following: * Serum monoclonal protein (serum protein electrophoresis \[SPEP\]) \> 1gm/dL * Serum free light chain (sFLC): involved free light chain (FLC) \>= 10mg/dL AND abnormal kappa to lambda serum free light chain ratio * \>= 200mg of monoclonal protein in the urine on 24 hour electrophoresis (urine protein electrophoresis \[UPEP\]) * Adequate bone marrow and major organ function to undergo a PBSC transplant determined within 30-60 days prior to enrollment using standard phase 1 criteria for organ function defined as: * Absolute neutrophil count (ANC) \>= 1.5 x 10\^9 cells/L * Platelets \>= 75 x 10\^9/L * Hemoglobin \>= 8 g/dL * Aspartate and alanine aminotransferases (AST, ALT) =\< 2.5 x upper limit of normal (ULN) (=\< 5 x ULN, if documented liver metastases are present) * Total bilirubin =\< 2 x ULN (except patients with documented Gilbert?s syndrome) * Creatinine \< 2 mg/dl (or a glomerular filtration rate \> 60) * Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0 or 1 * Must be willing and able to accept at least three leukapheresis procedures * Must be willing and able to undergo three research PET scans * Must be willing and able to provide written informed consent Exclusion Criteria: * Inability to purify \>= 2.5 x 10\^6 CD34-enriched cells/kg of patient weight from the pooled granulocyte-colony stimulating factor (G-CSF) mobilized leukapheresis products * Previous allogeneic transplant * Previously known hypersensitivity to any of the agents used in this study; known sensitivity to melphalan * Received systemic treatment for multiple myeloma, including immunotherapy, within 14 days prior to initiation of study procedures * Potential requirement for systemic corticosteroids or concurrent immunosuppressive drugs based on prior history or received systemic steroids within the last 2 weeks prior to enrollment (inhaled or topical steroids at standard doses are allowed) * Human immunodeficiency virus (HIV) seropositivity or other congenital or acquired immune deficiency state, which would increase the risk of opportunistic infections and other complications during chemotherapy-induced lymphodepletion; if there is a positive result in the infectious disease testing that was not previously known, the patient will be referred to their primary physician and/or infectious disease specialist * Hepatitis B or C seropositivity with evidence of ongoing liver damage, which would increase the likelihood of hepatic toxicities from the chemotherapy conditioning regimen and supportive treatments; if there is a positive result in the infectious disease testing that was not previously known, the patient will be referred to their primary physician and/or infectious disease specialist * Dementia or significantly altered mental status that would prohibit the understanding or rendering of informed consent and compliance with the requirements of this protocol * Known clinically active central nervous system (CNS) involvement; prior evidence of CNS involvement successfully treated with surgery or radiation therapy will not be exclusion for participation as long as they are deemed under control at the time of study enrollment and there are no neurological signs of potential CNS involvement * Pregnancy or breast-feeding; female patients must be surgically sterile or be postmenopausal for two years, or must agree to use effective contraception during the period of treatment and for 6 months afterwards; all female patients with reproductive potential must have a negative pregnancy test (serum/urine) within 14 days from starting the conditioning chemotherapy; the definition of effective contraception will be based on the judgment of the study investigators * Since IL-2 is administered following cell infusion: * Patients will be excluded if they have a history of clinically significant electrocardiogram (ECG) abnormalities, symptoms of cardiac ischemia with evidence of ischemia on a cardiac stress test (stress thallium, stress multigated acquisition \[MUGA\], dobutamine echocardiogram or other stress test) * Similarly, patients with a baseline left ventricular ejection fraction (LVEF) \< 45 percent (%) will be excluded * Patients with ECG results of any conduction delays (PR interval \> 200 ms, corrected QT \[QTC\] \> 480 ms), sinus bradycardia (resting heart rate \< 50 beats per minute), sinus tachycardia (heart rate \> 120 beats per minute) will be evaluated by a cardiologist prior to starting the trial; patients with any arrhythmias, including atrial fibrillation/atrial flutter, excessive ectopy (defined as \> 20 premature ventricular contractions \[PVCs\] per minute), ventricular tachycardia or 3rd degree heart block will be excluded from the study unless cleared by a cardiologist * Patients with pulmonary function test abnormalities as evidenced by a forced expiratory volume in 1 (FEV1) / forced vital capacity (FVC) \< 70% of predicted for normality will be excluded * Active or recent herpes simplex virus (HSV) infection or cytomegalovirus (CMV) based on symptoms with positive swab culture and/or positive Immunoglobulin M (IgM) screening, which would complicate the post-conditioning period