Elotuzumab, Pomalidomide, & Dexamethasone (Elo-Pom-Dex) With Second Autologous Stem Cell Transplantation for Relapsed Multiple Myeloma

Inclusion Criteria: * Histologically confirmed diagnosis of multiple myeloma. * Received prior autologous stem cell transplantation as first line therapy for multiple myeloma with subsequent disease relapse/progression. * Failed 1 or 2 lines of treatment for multiple myeloma. A line of treatment includes all therapy including induction, transplant, and maintenance administered in a sequence in the absence of relapse/progression. Once relapse/progression occurs and subsequently the anti-myeloma treatment is changed, a new line of treatment has begun. Local radiation or corticosteroids will not be considered treatment for multiple myeloma. * Received 2 to 6 cycles of induction therapy per standard of care prior to 2nd autologous stem cell transplantation * Received standard of care melphalan conditioning for 2nd autologous stem cell transplantation, is currently Day +80 to +120 following transplant, and is responding to therapy (partial response or better as compared to pre-induction assessment. * All US study participants must be registered into the mandatory POMALYST REMS® program and be willing and able to comply with the requirements of the POMALYST REMS® program. For Canadian sites, patients will followed according to the Pomalidomide pregnancy prevention program * Females of reproductive potential within the US must agree to adhere to the scheduled pregnancy testing as required in the POMALYST REMS® program. For Canadian sites, patients will followed according to the Pomalidomide pregnancy prevention program * At least 18 and no more than 75 years of age at enrollment. * Eastern Cooperative Oncology Group (ECOG) performance status ≤ 2 * Normal bone marrow and organ function as defined as ALL of the following: * Absolute neutrophil count ≥ 1000/mm\^3 * Platelets ≥ 75,000/mm\^3 (transfusions not permitted within 7 days of screening) * Total bilirubin ≤ 2.0 x institutional upper limit of normal (IULN) * AST(SGOT)/ALT(SGPT) ≤ 3.0 x IULN * Creatinine clearance ≥ 15 mL/min * Females of childbearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control, abstinence) prior to study entry through Day +100 visit. Should a woman become pregnant or suspect she is pregnant while participating in this study, she must inform her treating physician immediately. * Able to understand and willing to sign an Institutional Review Board (IRB) approved written informed consent document. Exclusion Criteria: * Refractory to elotuzumab and/or pomalidomide, defined as progressive disease or clinical relapse on therapy or within 60 days following completion of therapy. Prior exposure to elotuzumab and/or pomalidomide is allowed as long as patient is not refractory to these agents. * More than one prior transplant prior to study entry with the exception of tandem transplantation. Tandem transplantation is defined as two autologous stem cell transplants that occur within 9 months of one another, and the patient did not have disease progression in the period between the two transplants. * Presence of peripheral neuropathy ≥ grade 3 based on National Cancer Institute (NCI) Common Terminology Criteria for Adverse Events (CTCAE) v 4.0 * History of plasma cell leukemia or MM central nervous system (CNS) involvement. * Receiving renal replacement therapy, hemodialysis, or peritoneal dialysis. * Diagnosed with another concurrent malignancy requiring treatment. * Known HIV or active hepatitis A, B, or C. Antidoby testing not required for screening * Known hypersensitivity to pomalidomide, dexamethasone, or any excipients in elotuzumab, formulation, or recombinant protein * Receiving any other investigational agents within 14 days prior to enrollment. * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, or cardiac arrhythmia. * Pregnant and/or breastfeeding. Females of childbearing potential must have two negative pregnancy tests. The first test should be performed within 10-14 days of study entry, and the second test within 24 hours prior to prescribing pomalidomide.